Calculus Examples

$f (θ) = 9 \sin (10 \cos (θ))$

Step 1

Find the first derivative.

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Step 1.1

Since $9$ is constant with respect to $θ$ , the derivative of $9 \sin (10 \cos (θ))$ with respect to $θ$ is $9 \frac{d}{d θ} [\sin (10 \cos (θ))]$ .

$9 \frac{d}{d θ} [\sin (10 \cos (θ))]$

Step 1.2

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 $g (θ) = 10 \cos (θ)$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $10 \cos (θ)$ .

$9 (\frac{d}{d u} [\sin (u)] \frac{d}{d θ} [10 \cos (θ)])$

Step 1.2.2

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

$9 (\cos (u) \frac{d}{d θ} [10 \cos (θ)])$

Step 1.2.3

Replace all occurrences of $u$ with $10 \cos (θ)$ .

$9 (\cos (10 \cos (θ)) \frac{d}{d θ} [10 \cos (θ)])$

Step 1.3

Differentiate using the Constant Multiple Rule.

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Step 1.3.1

Since $10$ is constant with respect to $θ$ , the derivative of $10 \cos (θ)$ with respect to $θ$ is $10 \frac{d}{d θ} [\cos (θ)]$ .

$9 \cos (10 \cos (θ)) (10 \frac{d}{d θ} [\cos (θ)])$

Step 1.3.2

Multiply $10$ by $9$ .

$90 \cos (10 \cos (θ)) \frac{d}{d θ} [\cos (θ)]$

Step 1.4

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

$90 \cos (10 \cos (θ)) (- \sin (θ))$

Step 1.5

Multiply $- 1$ by $90$ .

$f' (θ) = - 90 \cos (10 \cos (θ)) \sin (θ)$

Step 2

Find the second derivative.

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Step 2.1

Since $- 90$ is constant with respect to $θ$ , the derivative of $- 90 \cos (10 \cos (θ)) \sin (θ)$ with respect to $θ$ is $- 90 \frac{d}{d θ} [\cos (10 \cos (θ)) \sin (θ)]$ .

$- 90 \frac{d}{d θ} [\cos (10 \cos (θ)) \sin (θ)]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d θ} [f (θ) g (θ)]$ is $f (θ) \frac{d}{d θ} [g (θ)] + g (θ) \frac{d}{d θ} [f (θ)]$ where $f (θ) = \cos (10 \cos (θ))$ and $g (θ) = \sin (θ)$ .

$- 90 (\cos (10 \cos (θ)) \frac{d}{d θ} [\sin (θ)] + \sin (θ) \frac{d}{d θ} [\cos (10 \cos (θ))])$

Step 2.3

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

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) \frac{d}{d θ} [\cos (10 \cos (θ))])$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \cos (θ)$ and $g (θ) = 10 \cos (θ)$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $10 \cos (θ)$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (\frac{d}{d u} [\cos (u)] \frac{d}{d θ} [10 \cos (θ)]))$

Step 2.4.2

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

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (- \sin (u) \frac{d}{d θ} [10 \cos (θ)]))$

Step 2.4.3

Replace all occurrences of $u$ with $10 \cos (θ)$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (- \sin (10 \cos (θ)) \frac{d}{d θ} [10 \cos (θ)]))$

Step 2.5

Differentiate using the Constant Multiple Rule.

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Step 2.5.1

Since $10$ is constant with respect to $θ$ , the derivative of $10 \cos (θ)$ with respect to $θ$ is $10 \frac{d}{d θ} [\cos (θ)]$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (- \sin (10 \cos (θ)) (10 \frac{d}{d θ} [\cos (θ)])))$

Step 2.5.2

Multiply $10$ by $- 1$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (- 10 \sin (10 \cos (θ)) \frac{d}{d θ} [\cos (θ)]))$

Step 2.6

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

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (- 10 \sin (10 \cos (θ)) (- \sin (θ))))$

Step 2.7

Multiply $- 1$ by $- 10$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + \sin (θ) (10 \sin (10 \cos (θ)) \sin (θ)))$

Step 2.8

Raise $\sin (θ)$ to the power of $1$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + 10 \sin (10 \cos (θ)) (\sin^{1} (θ) \sin (θ)))$

Step 2.9

Raise $\sin (θ)$ to the power of $1$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + 10 \sin (10 \cos (θ)) (\sin^{1} (θ) \sin^{1} (θ)))$

Step 2.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 90 (\cos (10 \cos (θ)) \cos (θ) + 10 \sin (10 \cos (θ)) {\sin (θ)}^{1 + 1})$

Step 2.11

Add $1$ and $1$ .

$- 90 (\cos (10 \cos (θ)) \cos (θ) + 10 \sin (10 \cos (θ)) \sin^{2} (θ))$

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

$- 90 (\cos (10 \cos (θ)) \cos (θ)) - 90 (10 \sin (10 \cos (θ)) \sin^{2} (θ))$

Step 2.12.2

Multiply $10$ by $- 90$ .

$- 90 \cos (10 \cos (θ)) \cos (θ) - 900 \sin (10 \cos (θ)) \sin^{2} (θ)$

Step 2.12.3

Reorder terms.

$f'' (θ) = - 90 \cos (θ) \cos (10 \cos (θ)) - 900 \sin^{2} (θ) \sin (10 \cos (θ))$

Step 3

The second derivative of $f (θ)$ with respect to $θ$ is $- 90 \cos (θ) \cos (10 \cos (θ)) - 900 \sin^{2} (θ) \sin (10 \cos (θ))$ .

$- 90 \cos (θ) \cos (10 \cos (θ)) - 900 \sin^{2} (θ) \sin (10 \cos (θ))$