Calculus Examples

$r = \sin (θ^{2}) \cos (6 θ)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d θ} (r) = \frac{d}{d θ} (\sin (θ^{2}) \cos (6 θ))$

Step 2

The derivative of $r$ with respect to $θ$ is $r'$ .

$r'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (θ) = \sin (θ^{2})$ and $g (θ) = \cos (6 θ)$ .

$\sin (θ^{2}) \frac{d}{d θ} [\cos (6 θ)] + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \cos (θ)$ and $g (θ) = 6 θ$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $6 θ$ .

$\sin (θ^{2}) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d θ} [6 θ]) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.2.2

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

$\sin (θ^{2}) (- \sin (u_{1}) \frac{d}{d θ} [6 θ]) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $6 θ$ .

$\sin (θ^{2}) (- \sin (6 θ) \frac{d}{d θ} [6 θ]) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.3

Differentiate.

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Step 3.3.1

Since $6$ is constant with respect to $θ$ , the derivative of $6 θ$ with respect to $θ$ is $6 \frac{d}{d θ} [θ]$ .

$\sin (θ^{2}) (- \sin (6 θ) (6 \frac{d}{d θ} [θ])) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.3.2

Multiply $6$ by $- 1$ .

$\sin (θ^{2}) (- 6 \sin (6 θ) \frac{d}{d θ} [θ]) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.3.3

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

$\sin (θ^{2}) (- 6 \sin (6 θ) \cdot 1) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.3.4

Multiply $- 6$ by $1$ .

$\sin (θ^{2}) (- 6 \sin (6 θ)) + \cos (6 θ) \frac{d}{d θ} [\sin (θ^{2})]$

Step 3.4

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

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $θ^{2}$ .

$\sin (θ^{2}) (- 6 \sin (6 θ)) + \cos (6 θ) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d θ} [θ^{2}])$

Step 3.4.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\sin (θ^{2}) (- 6 \sin (6 θ)) + \cos (6 θ) (\cos (u_{2}) \frac{d}{d θ} [θ^{2}])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $θ^{2}$ .

$\sin (θ^{2}) (- 6 \sin (6 θ)) + \cos (6 θ) (\cos (θ^{2}) \frac{d}{d θ} [θ^{2}])$

Step 3.5

Differentiate using the Power Rule.

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Step 3.5.1

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

$\sin (θ^{2}) (- 6 \sin (6 θ)) + \cos (6 θ) \cos (θ^{2}) (2 θ)$

Step 3.5.2

Reorder terms.

$- 6 \sin (θ^{2}) \sin (6 θ) + 2 θ \cos (θ^{2}) \cos (6 θ)$

Step 4

Reform the equation by setting the left side equal to the right side.

$r' = - 6 \sin (θ^{2}) \sin (6 θ) + 2 θ \cos (θ^{2}) \cos (6 θ)$

Step 5

Replace $r'$ with $\frac{d r}{d θ}$ .

$\frac{d r}{d θ} = - 6 \sin (θ^{2}) \sin (6 θ) + 2 θ \cos (θ^{2}) \cos (6 θ)$