Table Of Derivatives: Rules, Trig & Formulas

What Is A Table Of Derivatives?

A Table of Derivatives is a reference chart that pairs each standard function with its derivative. Instead of re-deriving $\frac{d}{dx}\sin x$ or $\frac{d}{dx}\ln x$ from the limit definition every time, you look the result up, the same way you once used a multiplication table. The table has two halves: the rules that tell you how to differentiate combinations of functions, and the standard derivatives of the named functions themselves.

Every entry answers the same question in two languages at once. Algebraically, the derivative is the formula in the right-hand column. Geometrically, that formula is the slope of the tangent line to the graph at each point $x$. The table of $\frac{d}{dx}\sin x = \cos x$, for instance, is really the statement that the steepness of a sine wave at any point equals the height of the matching cosine wave.

Throughout this article we use one consistent notation, the Leibniz operator $\frac{d}{dx}$, read as "the derivative with respect to $x$ of." Where an input is itself a function $u = g(x)$, the chain rule adds a factor of $\frac{du}{dx}$, and we give that chain-rule form beside every entry.

What Are The Basic Rules In The Table Of Derivatives?

The rules are the machinery. They let you differentiate any combination built from the standard functions. Learn these seven and the rest of the table becomes a set of ingredients you can freely combine.

Table: The differentiation rules, with the constant $c$, the exponent $n$, and functions $f$, $g$, $u = g(x)$.

The power rule is the workhorse, and it holds for any real exponent: negative powers ($x^{-2}$), roots written as fractions ($x^{1/2}$), and constants ($x^{0} = 1$, whose derivative is $0$) all obey it. The full set of combination rules lives in the rules of differentiation reference, and the chain rule earns its own detailed treatment because it appears in the chain-rule column of every family below.

What Are The Standard Function Derivatives?

These are the exponential and logarithmic entries, the first "named function" family. The distinction that matters most on this table is between a variable base with a constant exponent ($x^{n}$, handled by the power rule) and a constant base with a variable exponent ($a^{x}$, handled here). They look similar and behave completely differently.

Table: Exponential and logarithmic derivatives. Here $a > 0$ with $a \neq 1$, and $u = g(x)$.

The function $e^{x}$ is the one graph in mathematics whose slope at every point equals its own height, which is exactly why $\frac{d}{dx}e^{x} = e^{x}$. That single fact is the reason $e$ sits at the centre of growth, decay, and compound interest. The full derivations sit in the derivatives of exponential functions and derivatives of logarithmic functions articles.

What Are The Derivatives Of The Six Trigonometric Functions?

The trig family is the block students memorise first. Notice the sign pattern: the three "co-" functions ($\cos$, $\cot$, $\csc$) all pick up a minus sign when differentiated.

Table: Derivatives of the six trigonometric functions, with $u = g(x)$.

Every one of these holds only when $x$ is measured in radians. In degrees an extra factor of $\frac{\pi}{180}$ appears, which is why calculus is done in radians throughout. The step-by-step derivations from the limit $\lim_{h \to 0}\frac{\sin h}{h} = 1$ are worked in derivatives of trigonometric functions.

What Are The Derivatives Of The Six Inverse Trigonometric Functions?

The inverse trig family turns trig ratios back into angles, and its derivatives are algebraic fractions with no trig left in them. The same "co-" sign pattern appears: $\arccos$, $\text{arccot}$, and $\text{arccsc}$ carry the minus sign, mirroring their partners exactly.

Table: Derivatives of the six inverse trigonometric functions, with $u = g(x)$.

Because $\arcsin$ and $\arccos$ share the same derivative up to a sign, their graphs have tangent slopes that are exact mirror images, and their sum $\arcsin x + \arccos x = \frac{\pi}{2}$ is constant, whose derivative is $0$. The implicit-differentiation proofs are in derivatives of inverse trigonometric functions.

What Are The Derivatives Of The Hyperbolic Functions?

Hyperbolic functions are built from $e^{x}$ and $e^{-x}$: $\sinh x = \frac{e^{x} - e^{-x}}{2}$ and $\cosh x = \frac{e^{x} + e^{-x}}{2}$. Their derivatives look almost like the trig family, with one welcome difference: $\frac{d}{dx}\cosh x = +\sinh x$ carries no minus sign, unlike $\cos$.

Table: Derivatives of the hyperbolic functions, with $u = g(x)$.

You can check the first one straight from the definition: $\frac{d}{dx}\sinh x = \frac{d}{dx}\cdot\frac{e^{x} - e^{-x}}{2} = \frac{e^{x} + e^{-x}}{2} = \cosh x$. The wider family of identities sits in the hyperbolic functions reference.

How Do You Use The Table Of Derivatives? Worked Lookups

A table is only useful if you can combine its rows. Each lookup below picks the right rule, pulls the standard derivatives from the tables above, and finishes with a checked answer.

Example 1: Power rule, constant multiple, and sum together.

Differentiate $f(x) = 5x^{4} - 3x^{2} + 7$.

Apply the power rule term by term, keep the scalars, and send the constant to zero:

$$\frac{d}{dx}\big(5x^{4} - 3x^{2} + 7\big) = 5\cdot 4x^{3} - 3\cdot 2x^{1} + 0 = 20x^{3} - 6x$$

Final answer: $20x^{3} - 6x$.

Example 2: Product rule with an exponential row.

Differentiate $f(x) = x^{2}e^{x}$.

This is a product, so use $\frac{d}{dx}(fg) = f\frac{d}{dx}g + g\frac{d}{dx}f$ with $f = x^{2}$ and $g = e^{x}$:

$$\frac{d}{dx}\big(x^{2}e^{x}\big) = x^{2}\cdot e^{x} + e^{x}\cdot 2x = e^{x}\big(x^{2} + 2x\big)$$

Final answer: $e^{x}\big(x^{2} + 2x\big)$, or $x e^{x}(x + 2)$.

Example 3: Quotient rule with a trig row.

Differentiate $f(x) = \dfrac{\sin x}{x}$.

Use the quotient rule with $f = \sin x$ (derivative $\cos x$) and $g = x$ (derivative $1$):

$$\frac{d}{dx}\left(\frac{\sin x}{x}\right) = \frac{x\cos x - \sin x\cdot 1}{x^{2}} = \frac{x\cos x - \sin x}{x^{2}}$$

Final answer: $\dfrac{x\cos x - \sin x}{x^{2}}$.

Example 4: Chain rule with an inverse-trig row.

Differentiate $f(x) = \arctan(3x)$.

The outer function is $\arctan$, whose row gives $\frac{1}{1 + u^{2}}$; the inner is $u = 3x$ with $\frac{du}{dx} = 3$:

$$\frac{d}{dx}\arctan(3x) = \frac{1}{1 + (3x)^{2}}\cdot 3 = \frac{3}{1 + 9x^{2}}$$

Final answer: $\dfrac{3}{1 + 9x^{2}}$.

Example 5: Chain rule with a hyperbolic row.

Differentiate $f(x) = \cosh(5x)$.

The $\cosh$ row gives $\sinh u$ with no sign flip; the inner is $u = 5x$ with $\frac{du}{dx} = 5$:

$$\frac{d}{dx}\cosh(5x) = \sinh(5x)\cdot 5 = 5\sinh(5x)$$

Final answer: $5\sinh(5x)$.

Why Does Each Derivative In The Table Work?

None of these entries are arbitrary. Each is the slope of a tangent line, and the whole table is one geometric idea applied to different curves.

• A derivative is a slope. The value in the right-hand column is how steep the graph is at each $x$. For $\frac{d}{dx}x^{2} = 2x$, the parabola is flat at the bottom ($x = 0$ gives slope $0$) and steeper as you move out, which is exactly what $2x$ reports.

• The sign tells you the direction. A minus sign, as in $\frac{d}{dx}\cos x = -\sin x$, says the curve is falling where its partner is positive. The "co-" functions all fall where their partners rise, which is the whole reason for the shared minus pattern.

• $e^{x}$ is the fixed point of differentiation. It is the one non-zero function that equals its own derivative, so its tangent slope always matches its height. Every other exponential $a^{x}$ inherits an extra factor of $\ln a$ because it grows at a different rate than $e^{x}$.

• Chain rule accounts for inside speed. When the input is a function, the curve is being fed a value that is itself changing, so the outer slope is multiplied by how fast the inside moves, $\frac{du}{dx}$.

Read this way, memorising the table is really memorising a gallery of tangent-slope pictures. The definition of the derivative is where every row ultimately comes from.

Who Created The Table Of Derivatives?

The table did not arrive complete. Its rows were filled in over roughly two centuries, function family by function family.

Two more figures rounded out the table:

• Johann Heinrich Lambert (1728–1777, Switzerland/Germany) introduced the hyperbolic functions $\sinh$ and $\cosh$ as their own named family, giving the table its final block.

• Augustin-Louis Cauchy (1789–1857, France) put the derivative on the rigorous limit footing that every entry now rests on, so the table is a set of theorems rather than a set of lucky patterns.

Where Is The Table Of Derivatives Used In The Real World?

The table is the lookup layer under any calculation that involves a rate of change.

• Physics and engineering: velocity is the derivative of position and acceleration the derivative of velocity, so a single trig or polynomial row turns a motion equation into a rate in one step.

• Economics: marginal cost and marginal revenue are derivatives of total cost and total revenue, and the exponential rows model continuously compounded growth.

• Medicine and biology: drug concentration and population size follow exponential and logistic curves, and the $e^{x}$ row gives the instantaneous rate a dose decays or a colony grows.

• Machine learning: training a model means differentiating a loss function millions of times, and the chain-rule column is exactly the backpropagation step that makes it possible.

• Signal processing and graphics: the trig and hyperbolic rows describe oscillations and smooth curves, and their derivatives drive filtering, animation easing, and edge detection.

One chart of standard slopes quietly powers motion, money, medicine, and machines, which is why every quantitative field keeps a version of it close.

What Are The Most Common Mistakes With The Table Of Derivatives?

These three errors account for most lost marks when students read the table, and each matches a question real learners ask on Google's "People also ask" and r/calculus.

Using the power rule on an exponential.

Where it slips in:

A student sees $e^{x}$ or $2^{x}$ and copies the power-rule pattern, writing $\frac{d}{dx}e^{x} = x,e^{x-1}$ or $\frac{d}{dx}2^{x} = x,2^{x-1}$.

Don't do this:

Do not apply $n,x^{n-1}$ when the variable is in the exponent. The power rule is only for a variable base with a constant exponent, like $x^{n}$.

The correct way:

Use the exponential rows. $\frac{d}{dx}e^{x} = e^{x}$, and $\frac{d}{dx}a^{x} = a^{x}\ln a$, so $\frac{d}{dx}2^{x} = 2^{x}\ln 2$.

Losing the minus sign on $\cos$ or $\arccos$.

Where it slips in:

A student writes $\frac{d}{dx}\cos x = \sin x$ or $\frac{d}{dx}\arccos x = \dfrac{1}{\sqrt{1 - x^{2}}}$, dropping the negative that both "co-" entries carry.

Don't do this:

Do not assume a function and its derivative share a sign. Watch the "co-" pattern: $\cos$, $\cot$, $\csc$, $\arccos$, $\text{arccot}$, and $\text{arccsc}$ all pick up a minus.

The correct way:

Read the sign straight from the row: $\frac{d}{dx}\cos x = -\sin x$ and $\frac{d}{dx}\arccos x = -\dfrac{1}{\sqrt{1 - x^{2}}}$. Note the deliberate contrast with $\frac{d}{dx}\cosh x = +\sinh x$, which has no minus.

Forgetting the inner derivative in the chain-rule column.

Where it slips in:

A student differentiates $\sin(x^{2})$ as $\cos(x^{2})$ and stops, using the bare-$x$ column when the input is a function.

Don't do this:

Do not read across the "Derivative" column when the argument is not a plain $x$. That column assumes the input is $x$ itself.

The correct way:

Use the chain-rule column: $\frac{d}{dx}\sin(x^{2}) = \cos(x^{2})\cdot\frac{d}{dx}(x^{2}) = \cos(x^{2})\cdot 2x = 2x\cos(x^{2})$. Always multiply by the derivative of the inside.

Practice Problems On The Table Of Derivatives

Work each one, then check against the verified answer.

1. $\frac{d}{dx},x^{7}$.
   (Answer: $7x^{6}$.)

2. $\frac{d}{dx}\cos x$.
   (Answer: $-\sin x$.)

3. $\frac{d}{dx}\big(e^{x}\sin x\big)$ (product rule).
   (Answer: $e^{x}\sin x + e^{x}\cos x = e^{x}(\sin x + \cos x)$.)

4. $\frac{d}{dx}\ln\big(x^{2} + 1\big)$ (chain rule).
   (Answer: $\dfrac{2x}{x^{2} + 1}$.)

5. $\frac{d}{dx}\arcsin x$.
   (Answer: $\dfrac{1}{\sqrt{1 - x^{2}}}$.)

6. $\frac{d}{dx}\tanh(2x)$ (chain rule).
   (Answer: $2\operatorname{sech}^{2}(2x)$.)

Where Should You Go Next After The Table Of Derivatives?

The table is the launch pad for the whole differentiation toolkit, and several natural doors open from here.

1. Chain rule. Master the one rule that appears in every chain-rule column, since it unlocks composite functions across the entire table.

2. Derivative formulas. See each row derived and extended, with more combined worked examples than a reference chart can hold.

3. Table of integrals. Run the table in reverse to build the antiderivative lookup you need for area and accumulation problems.

If your child is learning to differentiate for the first time, a live Bhanzu trainer teaches the table from the tangent-slope picture up, so each family feels like one idea rather than a wall of formulas, in the Bhanzu math program.