Calculus Examples

$y = \sin (θ + \tan (θ + \cos (θ)))$

Step 1

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

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Step 1.1

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

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d θ} [θ + \tan (θ + \cos (θ))]$

Step 1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d θ} [θ + \tan (θ + \cos (θ))]$

Step 1.3

Replace all occurrences of $u_{1}$ with $θ + \tan (θ + \cos (θ))$ .

$\cos (θ + \tan (θ + \cos (θ))) \frac{d}{d θ} [θ + \tan (θ + \cos (θ))]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $θ + \tan (θ + \cos (θ))$ with respect to $θ$ is $\frac{d}{d θ} [θ] + \frac{d}{d θ} [\tan (θ + \cos (θ))]$ .

$\cos (θ + \tan (θ + \cos (θ))) (\frac{d}{d θ} [θ] + \frac{d}{d θ} [\tan (θ + \cos (θ))])$

Step 2.2

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

$\cos (θ + \tan (θ + \cos (θ))) (1 + \frac{d}{d θ} [\tan (θ + \cos (θ))])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \tan (θ)$ and $g (θ) = θ + \cos (θ)$ .

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Step 3.1

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

$\cos (θ + \tan (θ + \cos (θ))) (1 + \frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d θ} [θ + \cos (θ)])$

Step 3.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$\cos (θ + \tan (θ + \cos (θ))) (1 + \sec^{2} (u_{2}) \frac{d}{d θ} [θ + \cos (θ)])$

Step 3.3

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

$\cos (θ + \tan (θ + \cos (θ))) (1 + \sec^{2} (θ + \cos (θ)) \frac{d}{d θ} [θ + \cos (θ)])$

Step 4

Differentiate.

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Step 4.1

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

$\cos (θ + \tan (θ + \cos (θ))) (1 + \sec^{2} (θ + \cos (θ)) (\frac{d}{d θ} [θ] + \frac{d}{d θ} [\cos (θ)]))$

Step 4.2

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

$\cos (θ + \tan (θ + \cos (θ))) (1 + \sec^{2} (θ + \cos (θ)) (1 + \frac{d}{d θ} [\cos (θ)]))$

Step 5

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

$\cos (θ + \tan (θ + \cos (θ))) (1 + \sec^{2} (θ + \cos (θ)) (1 - \sin (θ)))$