Calculus Examples

$h (θ) = θ \cos (θ)$

Step 1

Find the first derivative.

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Step 1.1

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 $f (θ) = θ$ and $g (θ) = \cos (θ)$ .

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

$θ (- \sin (θ)) + \cos (θ) \cdot 1$

Step 1.3.2

Simplify the expression.

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Step 1.3.2.1

Multiply $\cos (θ)$ by $1$ .

$θ (- \sin (θ)) + \cos (θ)$

Step 1.3.2.2

Reorder terms.

$f' (θ) = - θ \sin (θ) + \cos (θ)$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d θ} [- θ \sin (θ)]$ .

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Step 2.2.1

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

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

Step 2.2.2

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 $f (θ) = θ$ and $g (θ) = \sin (θ)$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $\sin (θ)$ by $1$ .

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- (θ \cos (θ)) - \sin (θ) - \sin (θ)$

Step 2.4.2

Subtract $\sin (θ)$ from $- \sin (θ)$ .

$f'' (θ) = - θ \cos (θ) - 2 \sin (θ)$

Step 3

The second derivative of $h (θ)$ with respect to $θ$ is $- θ \cos (θ) - 2 \sin (θ)$ .

$- θ \cos (θ) - 2 \sin (θ)$