Calculus Examples

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

Step 1

Take the derivative of $\int_{0}^{x^{3}} f t d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4

Add $2$ and $3$ .

$3 (f x^{5})$

Step 5

Reorder the factors of $3 f x^{5}$ .

$3 x^{5} f$