Calculus Examples

\int_{0}^{x^{3}} f (t) d t = x^{4}

$\int_{0}^{x^{3}} f (t) d t = x^{4}$

Step 1

Take the derivative of

\int_{0}^{x^{3}} f t d t

$\int_{0}^{x^{3}} f t d t$ with respect to

x

$x$ using Fundamental Theorem of Calculus and the chain rule.

\frac{d}{d x} [x^{3}] (f x^{3})

$\frac{d}{d x} [x^{3}] (f x^{3})$

Step 2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

3 x^{2} (f x^{3})

$3 x^{2} (f x^{3})$

Step 3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

3 (f x^{2 + 3})

$3 (f x^{2 + 3})$

Step 4

Add

2

$2$ and

3

$3$ .

3 (f x^{5})

$3 (f x^{5})$

Step 5

Reorder the factors of

3 f x^{5}

$3 f x^{5}$ .

3 x^{5} f

$3 x^{5} f$