Calculus Examples

$y = 4 \sin^{2} (x) + 5 \cos^{2} (x)$

By the Sum Rule, the derivative of $4 \sin^{2} (x) + 5 \cos^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [4 \sin^{2} (x)] + \frac{d}{d x} [5 \cos^{2} (x)]$ .

$\frac{d}{d x} [4 \sin^{2} (x)] + \frac{d}{d x} [5 \cos^{2} (x)]$

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

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

$4 \frac{d}{d x} [\sin^{2} (x)] + \frac{d}{d x} [5 \cos^{2} (x)]$

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

$4 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [5 \cos^{2} (x)]$

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

$4 (2 u_{1} \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [5 \cos^{2} (x)]$

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

$4 (2 \sin (x) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [5 \cos^{2} (x)]$

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

$4 (2 \sin (x) \cos (x)) + \frac{d}{d x} [5 \cos^{2} (x)]$

Multiply $2$ by $4$ .

$8 \sin (x) \cos (x) + \frac{d}{d x} [5 \cos^{2} (x)]$

Evaluate $\frac{d}{d x} [5 \cos^{2} (x)]$ .

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Since $5$ is constant with respect to $x$ , the derivative of $5 \cos^{2} (x)$ with respect to $x$ is $5 \frac{d}{d x} [\cos^{2} (x)]$ .

$8 \sin (x) \cos (x) + 5 \frac{d}{d x} [\cos^{2} (x)]$

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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To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$8 \sin (x) \cos (x) + 5 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (x)])$

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

$8 \sin (x) \cos (x) + 5 (2 u_{2} \frac{d}{d x} [\cos (x)])$

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

$8 \sin (x) \cos (x) + 5 (2 \cos (x) \frac{d}{d x} [\cos (x)])$

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

$8 \sin (x) \cos (x) + 5 (2 \cos (x) (- \sin (x)))$

Multiply $- 1$ by $2$ .

$8 \sin (x) \cos (x) + 5 (- 2 \cos (x) \sin (x))$

Multiply $- 2$ by $5$ .

$8 \sin (x) \cos (x) - 10 \cos (x) \sin (x)$

Combine terms.

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Reorder the factors of $8 \sin (x) \cos (x)$ .

$8 \cos (x) \sin (x) - 10 \cos (x) \sin (x)$

Subtract $10 \cos (x) \sin (x)$ from $8 \cos (x) \sin (x)$ .

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