Calculus Examples

$f'' (x) = - 2 + 30 x - 12 x^{2}$

Step 1

The function $f' (x)$ can be found by evaluating the indefinite integral of the derivative $f'' (x)$ .

$f' (x) = \int f'' (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int - 2 d x + \int 30 x d x + \int - 12 x^{2} d x$

Step 3

Apply the constant rule.

$- 2 x + C + \int 30 x d x + \int - 12 x^{2} d x$

Step 4

Since $30$ is constant with respect to $x$ , move $30$ out of the integral.

$- 2 x + C + 30 \int x d x + \int - 12 x^{2} d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 2 x + C + 30 (\frac{1}{2} x^{2} + C) + \int - 12 x^{2} d x$

Step 6

Since $- 12$ is constant with respect to $x$ , move $- 12$ out of the integral.

$- 2 x + C + 30 (\frac{1}{2} x^{2} + C) - 12 \int x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- 2 x + C + 30 (\frac{1}{2} x^{2} + C) - 12 (\frac{1}{3} x^{3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$- 2 x + 15 x^{2} - 12 (\frac{1}{3} x^{3}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$- 2 x + 15 x^{2} - 12 \frac{x^{3}}{3} + C$

Step 8.2.2

Combine $- 12$ and $\frac{x^{3}}{3}$ .

$- 2 x + 15 x^{2} + \frac{- 12 x^{3}}{3} + C$

Step 8.2.3

Cancel the common factor of $- 12$ and $3$ .

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Step 8.2.3.1

Factor $3$ out of $- 12 x^{3}$ .

$- 2 x + 15 x^{2} + \frac{3 (- 4 x^{3})}{3} + C$

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $3$ out of $3$ .

$- 2 x + 15 x^{2} + \frac{3 (- 4 x^{3})}{3 (1)} + C$

Step 8.2.3.2.2

Cancel the common factor.

$- 2 x + 15 x^{2} + \frac{3 (- 4 x^{3})}{3 \cdot 1} + C$

Step 8.2.3.2.3

Rewrite the expression.

$- 2 x + 15 x^{2} + \frac{- 4 x^{3}}{1} + C$

Step 8.2.3.2.4

Divide $- 4 x^{3}$ by $1$ .

$- 2 x + 15 x^{2} - 4 x^{3} + C$

Step 9

The function $f'$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f' (x) = - 2 x + 15 x^{2} - 4 x^{3} + C$

Step 10

The function $f (x)$ can be found by evaluating the indefinite integral of the derivative $f' (x)$ .

$f (x) = \int f' (x) d x$

Step 11

Split the single integral into multiple integrals.

$\int - 2 x d x + \int 15 x^{2} d x + \int - 4 x^{3} d x + \int C d x$

Step 12

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$- 2 \int x d x + \int 15 x^{2} d x + \int - 4 x^{3} d x + \int C d x$

Step 13

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 2 (\frac{1}{2} x^{2} + C) + \int 15 x^{2} d x + \int - 4 x^{3} d x + \int C d x$

Step 14

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$- 2 (\frac{1}{2} x^{2} + C) + 15 \int x^{2} d x + \int - 4 x^{3} d x + \int C d x$

Step 15

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- 2 (\frac{1}{2} x^{2} + C) + 15 (\frac{1}{3} x^{3} + C) + \int - 4 x^{3} d x + \int C d x$

Step 16

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 2 (\frac{1}{2} x^{2} + C) + 15 (\frac{1}{3} x^{3} + C) - 4 \int x^{3} d x + \int C d x$

Step 17

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$- 2 (\frac{1}{2} x^{2} + C) + 15 (\frac{1}{3} x^{3} + C) - 4 (\frac{1}{4} x^{4} + C) + \int C d x$

Step 18

Apply the constant rule.

$- 2 (\frac{1}{2} x^{2} + C) + 15 (\frac{1}{3} x^{3} + C) - 4 (\frac{1}{4} x^{4} + C) + C x + C$

Step 19

Simplify.

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Step 19.1

Simplify.

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Step 19.1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$- 2 (\frac{x^{2}}{2} + C) + 15 (\frac{1}{3} x^{3} + C) - 4 (\frac{1}{4} x^{4} + C) + C x + C$

Step 19.1.2

Combine $\frac{1}{3}$ and $x^{3}$ .

$- 2 (\frac{x^{2}}{2} + C) + 15 (\frac{x^{3}}{3} + C) - 4 (\frac{1}{4} x^{4} + C) + C x + C$

Step 19.1.3

Combine $\frac{1}{4}$ and $x^{4}$ .

$- 2 (\frac{x^{2}}{2} + C) + 15 (\frac{x^{3}}{3} + C) - 4 (\frac{x^{4}}{4} + C) + C x + C$

Step 19.2

Simplify.

$- x^{2} + 5 x^{3} - x^{4} + C x + C$

Step 20

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f (x) = - x^{2} + 5 x^{3} - x^{4} + C x + C$