Calculus Examples

$\int_{π}^{2 x} \sin (t^{2}) d t$

Step 1

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

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

Step 2

Differentiate.

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Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

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

$2 \cdot 1 \sin ({(2 x)}^{2})$

Simplify with factoring out.

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Multiply $2$ by $1$ .

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

Factor $2$ out of $2 x$ .

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

Simplify the expression.

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Apply the product rule to $2 (x)$ .

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

Raise $2$ to the power of $2$ .

$2 \sin (4 x^{2})$