Calculus Examples

$\frac{d}{d x} (\int_{0}^{x^{3}} \ln (t^{2} + 1) d t)$

Step 1

Take the derivative of $\int_{0}^{x^{3}} \ln (t^{2} + 1) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [x^{3}] \ln ({(x^{3})}^{2} + 1)$

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 $n = 3$ .

$3 x^{2} \ln ({(x^{3})}^{2} + 1)$

Step 2.2

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

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Step 2.2.1

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

$3 x^{2} \ln (x^{3 \cdot 2} + 1)$

Step 2.2.2

Multiply $3$ by $2$ .

$3 x^{2} \ln (x^{6} + 1)$