Calculus Examples

$\frac{d}{d x} (\int_{0}^{x^{2}} \sin (t^{3}) d t)$

Step 1

Take the derivative of $\int_{0}^{x^{2}} \sin (t^{3}) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [x^{2}] \sin ({(x^{2})}^{3})$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x \sin ({(x^{2})}^{3})$

Step 2.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 x \sin (x^{2 \cdot 3})$

Step 2.2.2

Multiply $2$ by $3$ .

$2 x \sin (x^{6})$