Differential of a Function: Formula, Examples, Errors

What Is The Differential Of A Function?

The differential of a function $y = f(x)$ is the quantity $dy = f'(x),dx$: the derivative at $x$ multiplied by a chosen change $dx$ in the input. It tells you how far $y$ would move if the function kept changing at its current rate. For small $dx$, that estimate sits very close to the true change in $y$.

Formally, let $f$ be differentiable at $x$. The differential of the independent variable is $dx$, which can be any real number (in practice a small one). The differential of the dependent variable is

$$dy = f'(x),dx$$

Here is what each symbol means:

The differential is a function of two inputs, $x$ and $dx$, while the derivative is a function of $x$ alone. That small fact settles most of the confusion that follows.

How Do You Find The Differential Of A Function?

You find the differential of a function in two moves: differentiate, then multiply by $dx$. For $y = x^3$, the derivative is $3x^2$, so $dy = 3x^2,dx$. For $y = \sin x$, $dy = \cos x,dx$. Actually, that last line needs one condition stated out loud: it holds only with $x$ measured in radians, because the derivative of sine is $\cos x$ only in radians.

Every differentiation rule has a differential version, because each one is the derivative rule with $dx$ attached. Writing $u$ and $v$ for differentiable functions of $x$ and $c$ for a constant:

The product line is the product rule multiplied through by $dx$; the quotient line comes from the quotient rule the same way. The power line hides the chain rule: $du$ already carries the inner derivative, since $du = u'(x),dx$.

What Does The Differential Mean Geometrically?

Geometrically, the differential of a function is the rise of the tangent line over a run of $dx$. Stand at the point $(x, f(x))$ on the curve and draw the tangent line there. Move $dx$ to the right. The tangent line climbs $f'(x),dx$, which is $dy$, while the curve itself climbs $\Delta y$.

The two rises agree when the curve is a straight line. On a curve that bends, the tangent line peels away from the curve, and the gap between $dy$ and $\Delta y$ is exactly that peeling. The tangent line equations page builds the line itself; here, the line's only job is to give a cheap stand-in for the curve near one point.

Why Is The Differential Not Equal To The Actual Change?

The differential is not equal to the actual change because it follows the tangent line, and the tangent line ignores the bending of the curve. A Physics Forums thread asks this directly: why is $\Delta y$ not equal to $dy$, even for a tiny $dx$? They are never equal on a curved graph, but their gap shrinks much faster than $dx$ does.

Take $y = x^3$ at $x = 2$, where $f'(2) = 12$, so $dy = 12,dx$:

Each time $dx$ gets 10 times smaller, the gap gets about 100 times smaller. Expanding shows why:

$$\Delta y - dy = (2 + dx)^3 - 8 - 12,dx$$

$$\Delta y - dy = 6,(dx)^2 + (dx)^3$$

The leading piece, $6,(dx)^2$, equals $\frac{1}{2}f''(2),(dx)^2$, since the second derivative of $x^3$ at $2$ is $12$. That half-second-derivative term is the curvature correction; adding it gives the quadratic step of Taylor polynomials. The differential is the first-order piece, and it is accurate precisely because the leftovers are second order.

The same thread family asks how $\frac{dy}{dx}$, $\frac{\Delta y}{\Delta x}$ and $\frac{\partial y}{\partial x}$ differ. The ratio $\frac{\Delta y}{\Delta x}$ is the slope of a secant line through two points on the curve. The ratio of differentials, $\frac{dy}{dx}$, equals $f'(x)$ exactly for any nonzero $dx$, because $dy$ was defined as $f'(x),dx$; it is the slope of the tangent. The curly $\partial$ marks a partial derivative, used when $y$ depends on several inputs and only one of them changes.

How Do Differentials Estimate Absolute, Relative And Percentage Error?

Differentials estimate errors by treating a measurement mistake as $dx$ and reading the resulting output mistake as $dy$. Suppose a quantity $y = f(x)$ is computed from a measured $x$, and the measurement could be off by $dx$. Then the error in $y$ is approximately $dy = f'(x),dx$. This is called propagated error: the input's error travels through the formula.

Three measures describe that error, and Brainly threads ask for each one with an example:

• Absolute error: the size of the error itself, approximately $|dy|$. It carries the units of $y$.

• Relative error: the absolute error compared with the size of the quantity, $\frac{dy}{y}$. It has no units.

• Percentage error: the relative error written per hundred, $\frac{dy}{y} \times 100$.

So the difference between relative and percentage error is only the factor of 100. A relative error of $0.006$ and a percentage error of $0.6$ say the same thing.

One shortcut covers every power formula. If $y = k,x^n$, then $dy = n,k,x^{n-1},dx$, and dividing by $y$ gives

$$\frac{dy}{y} = n,\frac{dx}{x}$$

The relative error gets multiplied by the power. Volume of a sphere goes with $r^3$, so a 1% error in the radius becomes about a 3% error in volume. A pendulum's period goes with $L^{1/2}$, so a 2% error in length becomes about 1% in the period: the clock in the opening paragraph.

Is A Differential The Same As A Derivative?

No: a derivative is a rate, and a differential is a change built from that rate. A Quora thread asks for the difference between the differential and the derivative of a function, and the clearest answer is to compare what each one takes in and gives back.

The terminology deserves an honest note. The word "differential" carries at least three meanings across textbooks (a small number $dx$, Leibniz's infinitely small quantity, and the linear map of advanced courses), and it takes a while to notice which one a given book means. This article uses the first meaning throughout, because it is the one that produces numbers.

Where Do Differentials Go Beyond One Small Change?

Differentials extend in three directions that reappear later in calculus: several input variables, higher orders, and integration.

How Does The Total Differential Work With Several Variables?

When $z = f(x, y)$ depends on two inputs, each input contributes its own share of the change. The total differential adds them:

$$dz = \frac{\partial f}{\partial x},dx + \frac{\partial f}{\partial y},dy$$

Each partial derivative $\frac{\partial f}{\partial x}$ is the ordinary derivative with the other input held fixed. For a cylinder, $V = \pi r^2 h$, the total differential divided by $V$ gives $\frac{dV}{V} = 2,\frac{dr}{r} + \frac{dh}{h}$: radius errors count double, height errors count once.

What Are Higher-Order Differentials?

Applying the differential again, with $dx$ held fixed, gives $d^2y = f''(x),(dx)^2$. That is the curvature term from the gap table above, with the factor $\frac{1}{2}$ supplied when it enters an approximation.

Why Does $dx$ Appear Inside Integrals?

The $dx$ in an integral is the same differential, and the integration by substitution method uses it directly. In $\int 2x\cos\left(x^2\right),dx$, set $u = x^2$, so $du = 2x,dx$:

$$\int 2x\cos\left(x^2\right),dx = \int \cos u,du$$

$$\int \cos u,du = \sin\left(x^2\right) + C$$

Differentiating $\sin\left(x^2\right)$ returns $2x\cos\left(x^2\right)$, which confirms the result.

Examples Of The Differential Of A Function

The six examples below move from a straight computation to error propagation in a physical formula.

Example 1

Find the differential of $y = x^2\ln x$, then evaluate it at $x = 1$ with $dx = 0.2$.

Differentiate with the product rule:

$$f'(x) = 2x\ln x + x$$

Attach $dx$:

$$dy = \left(2x\ln x + x\right)dx$$

At $x = 1$, $\ln 1 = 0$:

$$dy = \left(0 + 1\right)(0.2)$$

$$dy = 0.2$$

Final answer: $dy = \left(2x\ln x + x\right)dx$, and $dy = 0.2$ at $x = 1$, $dx = 0.2$.

Example 2

Use differentials to estimate $\sin 31^\circ$.

The tempting set-up starts from $\sin 30^\circ = \frac{1}{2}$ with $dx = 1$, because the angle grew by one degree:

$$dy = \cos 30^\circ \times 1$$

$$\sin 31^\circ \approx 0.5 + 0.8660 = 1.3660$$

That answer is impossible, since sine never exceeds $1$. The break is the unit. The rule $\frac{d}{dx}\sin x = \cos x$ holds only when $x$ is in radians, so $dx$ must be in radians too: $1^\circ = \frac{\pi}{180} \approx 0.017453$.

Redo it with $x = \frac{\pi}{6}$ and $dx = \frac{\pi}{180}$:

$$dy = \cos\frac{\pi}{6}\cdot\frac{\pi}{180}$$

$$dy \approx 0.86603 \times 0.017453$$

$$dy \approx 0.015115$$

$$\sin 31^\circ \approx 0.5 + 0.015115 = 0.5151$$

A calculator gives $0.5150$, so the estimate is off by less than $0.0001$.

Final answer: $\sin 31^\circ \approx 0.5151$.

Example 3

For $y = x^3$ at $x = 2$ with $dx = 0.1$, compare $dy$ with $\Delta y$.

The differential:

$$dy = 3x^2,dx = 3(4)(0.1)$$

$$dy = 1.2$$

The actual change:

$$\Delta y = (2.1)^3 - 2^3$$

$$\Delta y = 9.261 - 8 = 1.261$$

The gap is $0.061$, which matches $6,(dx)^2 + (dx)^3 = 0.06 + 0.001$.

Final answer: $dy = 1.2$, $\Delta y = 1.261$, and the differential underestimates by $0.061$.

Example 4

Approximate $\sqrt[3]{26}$ using differentials.

Choose $f(x) = x^{1/3}$ at $x = 27$, because $\sqrt[3]{27} = 3$ is exact. Here $dx = -1$, and the negative sign matters.

$$f'(x) = \frac{1}{3}x^{-2/3}$$

$$f'(27) = \frac{1}{3}\cdot\frac{1}{9} = \frac{1}{27}$$

$$dy = \frac{1}{27}\times(-1) \approx -0.0370$$

$$\sqrt[3]{26} \approx 3 - 0.0370 = 2.9630$$

The true value is $2.9625$. The first instinct here is to start from $x = 26$ and step forward; the step only helps when it starts from a value you already know exactly.

Final answer: $\sqrt[3]{26} \approx 2.9630$.

Example 5

A sphere's radius is measured as $10$ cm with a possible error of $0.02$ cm. Estimate the error in its volume and the percentage error.

Volume and its differential:

$$V = \frac{4}{3}\pi r^3$$

$$dV = 4\pi r^2,dr$$

$$dV = 4\pi(100)(0.02) = 8\pi \approx 25.13,\text{cm}^3$$

Relative and percentage error, using the power shortcut:

$$\frac{dV}{V} = 3,\frac{dr}{r} = 3 \times \frac{0.02}{10} = 0.006$$

Final answer: the volume may be off by about $25.13,\text{cm}^3$, a percentage error of $0.6$.

Example 6

A pendulum's period is $T = 2\pi\sqrt{\frac{L}{g}}$. If $L = 1$ m is measured with a 2% error, estimate the error in $T$ (take $g = 9.81,\text{m/s}^2$).

$T$ is a constant times $L^{1/2}$, so the relative errors follow the power rule:

$$\frac{dT}{T} = \frac{1}{2},\frac{dL}{L} = \frac{1}{2}(0.02) = 0.01$$

The period itself:

$$T = 2\pi\sqrt{\frac{1}{9.81}} \approx 2.0061,\text{s}$$

$$dT \approx 0.01 \times 2.0061 \approx 0.0201,\text{s}$$

The exact change is $0.0200,\text{s}$, so the differential is reliable to three decimal places. Over the $86400$ seconds of a day, a 1% longer period puts the clock about 14 minutes behind.

Final answer: about a 1% error in the period, roughly $0.0201$ s per swing.

Why Did Mathematicians Need The Differential?

The differential exists to answer one practical question: "how much does the output move when the input moves a little?" Before there was any formal theory of limits, that question already drove three kinds of work:

• Tangents and extremes. Gottfried Leibniz's 1684 paper introduced the $d$ notation in a method for finding tangents, maxima and minima. Treating $dy$ and $dx$ as small changes let him read a slope as their ratio.

• Measurement you can trust. Every laboratory number has an uncertainty. The propagation of error method in the NIST/SEMATECH e-Handbook of Statistical Methods multiplies each input's uncertainty by a derivative, called a sensitivity coefficient: the multi-input version of $dy = f'(x),dx$.

• Numbers without a calculator. Estimates like $\sqrt[3]{26}$ or $\sin 31^\circ$ from a nearby known value were how tables and engineering hand-calculations got done.

The differential also became the bookkeeping device of calculus. The history of calculus is partly the story of how $dx$ went from a mysterious infinitely small quantity to an ordinary number with a precise job.

The Mathematicians Behind The Differential Of A Function

Gottfried Wilhelm Leibniz (1646 to 1716, Germany) first wrote the integral sign and $dx$ in a manuscript dated 21 November 1675. His 1684 paper Nova Methodus pro Maximis et Minimis in Acta Eruditorum published the $d$ notation together with the rules for differentiating powers, products and quotients.

Augustin-Louis Cauchy (1789 to 1857, France) wrote the Cours d'analyse of 1821 to develop the basic theorems of calculus as rigorously as possible. The limit-based approach that grew from that work is what allows $dx$ to be an ordinary number and $dy = f'(x),dx$ to be an exact definition instead of an infinitely small leap of faith.

Where Are Differentials Used In Real Life?

Differentials are used wherever a small, known change in one quantity has to be turned into a change in another:

• Manufacturing tolerances. A machinist who can hold a radius to within $0.02$ mm uses $dV = 4\pi r^2,dr$ or its cousins to decide whether the resulting volume or mass stays in specification.

• Laboratory error bars. Physics and chemistry students report results such as $g$ from a pendulum with an uncertainty computed exactly as in Example 6.

• Timekeeping. Clockmakers adjust pendulum length knowing that a fractional change in length makes half that fractional change in the period.

• Linked rates. Dividing a differential relationship by $dt$ turns it into a rate equation, which is how related rates problems are set up.

• Sensitivity checks. Engineers ask which input a design is most sensitive to by comparing the coefficients in a total differential.

Where Solutions Go Off The Rails

Most marks lost on differentials come from four habits, and each one is easy to catch once you know to look for it.

Mistake 1: Measuring $dx$ In Degrees

Where it slips in:

Any differential of a trig function, such as estimating $\sin 31^\circ$ or $\cos 59^\circ$.

Don't do this:

Writing $dx = 1$ for "one degree" and getting $\sin 31^\circ \approx 1.366$.

The correct way:

Convert first: $dx = \frac{\pi}{180}$. The rusher who skips this check never notices that the answer exceeds $1$; a two-second "can sine be bigger than 1?" catches it.

Mistake 2: Treating $dy$ As The Exact Change

Where it slips in:

Questions that ask for the actual change, or for the error in an approximation.

Don't do this:

Reporting $dy = 1.2$ as the change in $x^3$ from $2$ to $2.1$.

The correct way:

Use $dy$ when the question says "approximate" or "estimate", and compute $\Delta y = f(x + dx) - f(x)$ when it asks for the actual change. The memorizer who learned "$dy$ equals the change" gets every comparison question wrong, because the whole point of those questions is the gap.

Mistake 3: Stopping At The Absolute Error

Where it slips in:

Percentage-error questions on volumes, areas and periods.

Don't do this:

Answering "the percentage error is $25.13$" for the sphere in Example 5.

The correct way:

Divide by the quantity itself before multiplying by 100: $\frac{dV}{V} \times 100$. With power formulas, skip the division entirely and multiply the input's relative error by the exponent. Students who learn the $n,\frac{dx}{x}$ shortcut stop mixing up absolute and relative error, because the shortcut only ever produces a relative error.

Mistake 4: Dropping The $dx$

Where it slips in:

"Find the differential" questions, especially right after a block of "find the derivative" questions.

Don't do this:

Writing $dy = 2x\ln x + x$.

The correct way:

Keep the $dx$: $dy = \left(2x\ln x + x\right)dx$. Without it, the line states that a change equals a slope, which mixes units. The student who drops $dx$ here is usually the same student who later drops it from an integral and then cannot see what to substitute.

Practice Problems On The Differential Of A Function

Work each one, then compare with the answer. Every answer has been recomputed.

1. Find $dy$ for $y = e^{3x}\sin x$.

2. Use differentials to approximate $\sqrt{49.5}$ to four decimal places.

3. The side of a cube is measured as $10$ cm with a possible error of $0.05$ cm. Estimate the error in its volume and the percentage error.

4. For $y = x^2 - 4x$ at $x = 5$ with $dx = 0.05$, find $dy$ and $\Delta y$.

5. The radius of a circle increases from $4$ to $4.1$. Use differentials to estimate the change in its area.

6. The radius of a sphere is measured with a 1% error. Estimate the percentage error in its surface area.

Answer to Question 1: $dy = e^{3x}\left(3\sin x + \cos x\right)dx$.

Answer to Question 2: $\sqrt{49.5} \approx 7 + \frac{0.5}{14} \approx 7.0357$ (the true value is $7.0356$).

Answer to Question 3: $dV = 3s^2,ds = 15,\text{cm}^3$, a percentage error of $1.5$.

Answer to Question 4: $dy = 0.3$ and $\Delta y = 0.3025$.

Answer to Question 5: $dA = 2\pi r,dr = 0.8\pi \approx 2.5133$ square units.

Answer to Question 6: surface area goes with $r^2$, so the error is about $2$.

What To Remember About The Differential Of A Function

• The differential of a function is $dy = f'(x),dx$, the rise of the tangent line over a run of $dx$.

• The actual change $\Delta y$ follows the curve, and the gap $\Delta y - dy$ shrinks like $(dx)^2$.

• Absolute error is about $|dy|$, relative error is $\frac{dy}{y}$, and percentage error is that ratio times 100.

• For power formulas, the relative error of the output is the exponent times the relative error of the input.

• Trig differentials need $dx$ in radians, never degrees.

Your Next Move

Work through Practice Problems 1 to 6; if Problem 3 or Problem 6 trips you up, return to the section on absolute, relative and percentage error, and if Problem 4 does, reread the table comparing $dy$ with $\Delta y$.

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Read More

• Differentiability of a function: the condition $dy = f'(x),dx$ needs before it means anything.

• Mean value theorem: the exact statement behind "the actual change equals a slope times $dx$".

• Newton's method: the tangent-line idea used to find roots instead of values.

• Table of derivatives: every $f'(x)$ you need to write a differential.

• Implicit differentiation: differentials applied to equations that are not solved for $y$.