Calculus Examples

$r = \sin (θ) \tan (θ)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d θ} (r) = \frac{d}{d θ} (\sin (θ) \tan (θ))$

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

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

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Rewrite $\sec (θ)$ in terms of sines and cosines.

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

Step 3.4.2.2

Apply the product rule to $\frac{1}{\cos (θ)}$ .

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

Step 3.4.2.3

One to any power is one.

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

Step 3.4.2.4

Combine $\frac{1}{\cos^{2} (θ)}$ and $\sin (θ)$ .

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

Step 3.4.2.5

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.4.2.5.1

Reorder $\cos (θ)$ and $\tan (θ)$ .

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

Step 3.4.2.5.2

Rewrite $\cos (θ) \tan (θ)$ in terms of sines and cosines.

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

Step 3.4.2.5.3

Cancel the common factors.

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

Step 3.4.3

Simplify each term.

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Step 3.4.3.1

Factor $\cos (θ)$ out of $\cos^{2} (θ)$ .

$\frac{\sin (θ)}{\cos (θ) \cos (θ)} + \sin (θ)$

Step 3.4.3.2

Separate fractions.

$\frac{1}{\cos (θ)} \cdot \frac{\sin (θ)}{\cos (θ)} + \sin (θ)$

Step 3.4.3.3

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$\frac{1}{\cos (θ)} \tan (θ) + \sin (θ)$

Step 3.4.3.4

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$\sec (θ) \tan (θ) + \sin (θ)$

Step 4

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

$r' = \sec (θ) \tan (θ) + \sin (θ)$

Step 5

Replace $r'$ with $\frac{d r}{d θ}$ .

$\frac{d r}{d θ} = \sec (θ) \tan (θ) + \sin (θ)$