Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\cos^{2} (x)]$ .

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Step 2.1

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 $g (x) = \cos (x)$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

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

Step 2.2

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

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

Step 2.3

Multiply $- 1$ by $2$ .

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

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin^{2} (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin^{2} (x)]$ .

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

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 $u_{2}$ as $\sin (x)$ .

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

Step 3.2.2

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

$- 2 \cos (x) \sin (x) - (2 u_{2} \frac{d}{d x} [\sin (x)])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

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

Step 3.3

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

$- 2 \cos (x) \sin (x) - (2 \sin (x) \cos (x))$

Step 3.4

Multiply $2$ by $- 1$ .

$- 2 \cos (x) \sin (x) - 2 \sin (x) \cos (x)$

Step 4

Combine terms.

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Step 4.1

Reorder the factors of $- 2 \sin (x) \cos (x)$ .

$- 2 \cos (x) \sin (x) - 2 \cos (x) \sin (x)$

Step 4.2

Subtract $2 \cos (x) \sin (x)$ from $- 2 \cos (x) \sin (x)$ .

$- 4 \cos (x) \sin (x)$