Calculus Examples

$y = \cos^{2} (θ)$

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

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

$\frac{d}{d u} [u^{2}] \frac{d}{d θ} [\cos (θ)]$

Step 1.1.2

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

$2 u \frac{d}{d θ} [\cos (θ)]$

Step 1.1.3

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

$2 \cos (θ) \frac{d}{d θ} [\cos (θ)]$

Step 1.2

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

$2 \cos (θ) (- \sin (θ))$

Step 1.3

Multiply $- 1$ by $2$ .

$f' (θ) = - 2 \cos (θ) \sin (θ)$

Step 2

Find the second derivative.

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Step 2.1

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

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

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 $f (θ) = \cos (θ)$ and $g (θ) = \sin (θ)$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Add $1$ and $1$ .

$- 2 (\cos^{2} (θ) - \sin^{2} (θ))$

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Multiply $- 1$ by $- 2$ .

$f'' (θ) = - 2 \cos^{2} (θ) + 2 \sin^{2} (θ)$