Calculus Examples

$\frac{6 x}{π} - 4 \sin^{2} (x)$

Step 1

By the Sum Rule, the derivative of $\frac{6 x}{π} - 4 \sin^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{6 x}{π}] + \frac{d}{d x} [- 4 \sin^{2} (x)]$ .

$\frac{d}{d x} [\frac{6 x}{π}] + \frac{d}{d x} [- 4 \sin^{2} (x)]$

Step 2

Evaluate $\frac{d}{d x} [\frac{6 x}{π}]$ .

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Step 2.1

Since $\frac{6}{π}$ is constant with respect to $x$ , the derivative of $\frac{6 x}{π}$ with respect to $x$ is $\frac{6}{π} \frac{d}{d x} [x]$ .

$\frac{6}{π} \frac{d}{d x} [x] + \frac{d}{d x} [- 4 \sin^{2} (x)]$

Step 2.2

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

$\frac{6}{π} \cdot 1 + \frac{d}{d x} [- 4 \sin^{2} (x)]$

Step 2.3

Multiply $\frac{6}{π}$ by $1$ .

$\frac{6}{π} + \frac{d}{d x} [- 4 \sin^{2} (x)]$

Step 3

Evaluate $\frac{d}{d x} [- 4 \sin^{2} (x)]$ .

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Step 3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \sin^{2} (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\sin^{2} (x)]$ .

$\frac{6}{π} - 4 \frac{d}{d x} [\sin^{2} (x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

$\frac{6}{π} - 4 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sin (x)])$

Step 3.2.2

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

$\frac{6}{π} - 4 (2 u \frac{d}{d x} [\sin (x)])$

Step 3.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

$\frac{6}{π} - 4 (2 \sin (x) \frac{d}{d x} [\sin (x)])$

Step 3.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{6}{π} - 4 (2 \sin (x) \cos (x))$

Step 3.4

Multiply $2$ by $- 4$ .

$\frac{6}{π} - 8 \sin (x) \cos (x)$

Step 4

Reorder terms.

$- 8 \cos (x) \sin (x) + \frac{6}{π}$