Calculus Examples

$\int_{0}^{x^{2}} \cos (t^{2} + 1) d t$

Step 1

Take the derivative of $\int_{0}^{x^{2}} \cos (t^{2} + 1) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [x^{2}] \cos ({(x^{2})}^{2} + 1)$

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 $n = 2$ .

$2 x \cos ({(x^{2})}^{2} + 1)$

Step 2.2

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

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Step 2.2.1

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

$2 x \cos (x^{2 \cdot 2} + 1)$

Step 2.2.2

Multiply $2$ by $2$ .

$2 x \cos (x^{4} + 1)$