Maths Formula for Class 12 with Examples

Key Points

The maths formula for class 12 set is built around calculus, with derivatives and integrals accounting for most of it.
Differentiation and integration are inverse operations, so each derivative in the list doubles as an integral read backwards.
Vectors and three dimensional geometry extend class 11 coordinate work from the plane into space.
Probability brings conditional probability and the multiplication rule, which depend on the class 11 counting formulas.
Nearly every class 12 formula has a class 11 prerequisite, which is why the two years are best revised together.

What Formulas Does Class 12 Maths Cover?

The maths formula for class 12 list is dominated by calculus. Five groups cover it: derivatives, integrals, applications of calculus, vectors and three dimensional geometry, and probability.

The calculus groups are the largest, but they are also the most systematic, because the two halves mirror each other.

Derivative Formulas

Function	Derivative
$x^n$	$nx^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$

Rule	Formula
Product	$(uv)' = u'v + uv'$
Quotient	$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
Chain	$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

The derivative formula page covers the basics and uv differentiation handles the product rule in detail.

Integral Formulas

Function	Integral
$x^n$	$\frac{x^{n+1}}{n+1} + C$ , $n \neq -1$
$\frac{1}{x}$	$\ln\lvert x \rvert + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
By parts	$\int uv' = uv - \int u'v$

Every row is a derivative row reversed, which is why the two tables should be learned as one. The integration formula and integration of uv pages cover these.

Vectors and Three Dimensional Geometry

Quantity	Formula
Magnitude	$\lvert \vec{a} \rvert = \sqrt{x^2+y^2+z^2}$
Dot product	$\vec{a} \cdot \vec{b} = \lvert a \rvert \lvert b \rvert \cos\theta$
Cross product magnitude	$\lvert \vec{a} \times \vec{b} \rvert = \lvert a \rvert \lvert b \rvert \sin\theta$
Angle between vectors	$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{\lvert a \rvert \lvert b \rvert}$
Distance between points	$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

The dot product uses cosine and the cross product uses sine, which is the distinction worth fixing early. The distance formula extends directly into three dimensions.

Probability

Quantity	Formula
Conditional	$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$
Multiplication	$P(A \cap B) = P(A)P(B \mid A)$
Bayes	$P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}$
Expected value	$E(X) = \sum x P(x)$

The conditional probability formula and expected value formula pages carry these in full.

Examples of Maths Formula for Class 12

Example 1

Differentiate $y = x^5$ .

$\frac{dy}{dx} = 5x^4$

Final answer: $5x^4$

Example 2

Integrate $\int x^5 , dx$ by reversing the derivative rule as $5x^4$ .

Since differentiating $x^5$ gives $5x^4$ , reversing it might seem to turn $x^5$ into $5x^4$ .

That runs the rule the wrong way. Integration raises the power rather than lowering it, so the exponent must go from 5 to 6, and the new power divides rather than multiplies.

$\int x^5 , dx = \frac{x^6}{6} + C$

Final answer: $\frac{x^6}{6} + C$ . Differentiating the answer returns $x^5$ , which is the check.

Example 3

Differentiate $y = x^2 \sin x$ .

Use the product rule with $u = x^2$ and $v = \sin x$ :

$\frac{dy}{dx} = 2x\sin x + x^2\cos x$

Final answer: $2x\sin x + x^2\cos x$

Example 4

Find the angle between $\vec{a} = (1,2,2)$ and $\vec{b} = (2,2,1)$ .

$\vec{a} \cdot \vec{b} = 2+4+2 = 8, \qquad \lvert a \rvert = \lvert b \rvert = 3$ $\cos\theta = \frac{8}{9}, \qquad \theta \approx 27.27^\circ$

Final answer: about $27.27^\circ$

Example 5

$P(A) = 0.5$ and $P(A \cap B) = 0.2$ . Find $P(B \mid A)$ .

$P(B \mid A) = \frac{0.2}{0.5} = 0.4$

Final answer: 0.4

Where Class 12 Students Trip Up

Mistake 1: Running the power rule backwards incorrectly

Where it slips in: Integration is taught as the reverse of differentiation, so the rule gets reversed loosely.

Don't do this: $\int x^n , dx = nx^{n-1}$

The correct way: $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$ . Differentiation lowers the power and multiplies; integration raises it and divides.

Students who differentiate their own integral before moving on catch every error of this kind, because the check returns the original function or it does not. It costs one line and is available on every integration question.

Mistake 2: Omitting the constant of integration

Where it slips in: The antiderivative looks like a complete answer.

Don't do this: $\int \cos x , dx = \sin x$

The correct way: $\sin x + C$ . Differentiating destroys any constant, so reversing the operation cannot recover which one was there, and the family of answers must be written.

Mistake 3: Swapping cosine and sine between the two vector products

Where it slips in: Both products involve two vectors and an angle.

Don't do this: Use sine in the dot product.

The correct way: The dot product takes cosine and the cross product takes sine. The dot product of perpendicular vectors is zero, which is a quick way to recall which is which.

The Short Version

The maths formula for class 12 set covers derivatives, integrals, vectors, three dimensional geometry and probability.
The derivative and integral tables are one table read in two directions, which halves the learning.
Integration raises the power and divides, while differentiation lowers it and multiplies.
The dot product uses cosine and the cross product uses sine, and perpendicular vectors give a zero dot product.
To work through the class 12 syllabus with a teacher, explore a high school math tutor or math classes online.

Practice These Three Before Moving On

Differentiate $y = x^3\cos x$ . Then integrate $\int (3x^2 + e^x),dx$ , and find the dot product of $(2,1,3)$ and $(1,4,2)$ .

Want a live Bhanzu trainer to walk through the class 12 formulas? Book a free demo class.

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Frequently Asked Questions

What is the full maths formula for class 12 list?

Derivatives and their rules, standard integrals and integration by parts, applications of calculus, vectors and three dimensional geometry, and probability including Bayes' theorem.

Which maths formula for class 12 group is largest?

Calculus, but the derivative and integral tables mirror each other, so learning one direction gives much of the other.

Do I need class 11 formulas for class 12?

Yes, directly. Trigonometric identities, sequences, complex numbers and the counting formulas are all assumed, which is why the two years revise well together.

Why does every integral carry a constant?

Because differentiating removes any constant term, so an antiderivative is only determined up to that constant.

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