Calculus Examples

$y = \sec (θ) \tan (θ)$

Step 1

Let $y = f (θ)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (θ))$ .

$\ln (y) = \ln (\sec (θ) \tan (θ))$

Step 2

Rewrite $\ln (\sec (θ) \tan (θ))$ as $\ln (\sec (θ)) + \ln (\tan (θ))$ .

$\ln (y) = \ln (\sec (θ)) + \ln (\tan (θ))$

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

Tap for more steps...

Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

$\frac{y'}{y} = \ln (\sec (θ)) + \ln (\tan (θ))$

Step 3.2

Differentiate the right hand side.

Tap for more steps...

Step 3.2.1

Differentiate $\ln (\sec (θ)) + \ln (\tan (θ))$ .

$\frac{y'}{y} = \frac{d}{d θ} [\ln (\sec (θ)) + \ln (\tan (θ))]$

Step 3.2.2

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

$\frac{y'}{y} = \frac{d}{d θ} [\ln (\sec (θ))] + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3

Evaluate $\frac{d}{d θ} [\ln (\sec (θ))]$ .

Tap for more steps...

Step 3.2.3.1

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

Tap for more steps...

Step 3.2.3.1.1

To apply the Chain Rule, set $u_{1}$ as $\sec (θ)$ .

$\frac{y'}{y} = \frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d θ} [\sec (θ)] + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3.1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{y'}{y} = \frac{1}{u_{1}} \frac{d}{d θ} [\sec (θ)] + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3.1.3

Replace all occurrences of $u_{1}$ with $\sec (θ)$ .

$\frac{y'}{y} = \frac{1}{\sec (θ)} \frac{d}{d θ} [\sec (θ)] + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3.2

The derivative of $\sec (θ)$ with respect to $θ$ is $\sec (θ) \tan (θ)$ .

$\frac{y'}{y} = \frac{1}{\sec (θ)} (\sec (θ) \tan (θ)) + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3.3

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

$\frac{y'}{y} = \frac{1}{\frac{1}{\cos (θ)}} (\sec (θ) \tan (θ)) + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (θ)}$ .

$\frac{y'}{y} = 1 \cos (θ) (\sec (θ) \tan (θ)) + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.3.5

Multiply $\cos (θ)$ by $1$ .

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{d}{d θ} [\ln (\tan (θ))]$

Step 3.2.4

Evaluate $\frac{d}{d θ} [\ln (\tan (θ))]$ .

Tap for more steps...

Step 3.2.4.1

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

Tap for more steps...

Step 3.2.4.1.1

To apply the Chain Rule, set $u_{2}$ as $\tan (θ)$ .

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d θ} [\tan (θ)]$

Step 3.2.4.1.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{1}{u_{2}} \frac{d}{d θ} [\tan (θ)]$

Step 3.2.4.1.3

Replace all occurrences of $u_{2}$ with $\tan (θ)$ .

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{1}{\tan (θ)} \frac{d}{d θ} [\tan (θ)]$

Step 3.2.4.2

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

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{1}{\tan (θ)} \sec^{2} (θ)$

Step 3.2.4.3

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

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{1}{\frac{\sin (θ)}{\cos (θ)}} \sec^{2} (θ)$

Step 3.2.4.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (θ)}{\cos (θ)}$ .

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \frac{\cos (θ)}{\sin (θ)} \sec^{2} (θ)$

Step 3.2.4.5

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

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \cot (θ) \sec^{2} (θ)$

Step 3.2.5

Simplify.

Tap for more steps...

Step 3.2.5.1

Reorder terms.

$\frac{y'}{y} = \cos (θ) \sec (θ) \tan (θ) + \sec^{2} (θ) \cot (θ)$

Step 3.2.5.2

Simplify each term.

Tap for more steps...

Step 3.2.5.2.1

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

Tap for more steps...

Step 3.2.5.2.1.1

Reorder $\cos (θ)$ and $\sec (θ)$ .

$\frac{y'}{y} = \sec (θ) \cos (θ) \tan (θ) + \sec^{2} (θ) \cot (θ)$

Step 3.2.5.2.1.2

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

$\frac{y'}{y} = \frac{1}{\cos (θ)} \cos (θ) \tan (θ) + \sec^{2} (θ) \cot (θ)$

Step 3.2.5.2.1.3

Cancel the common factors.

$\frac{y'}{y} = 1 \tan (θ) + \sec^{2} (θ) \cot (θ)$

Step 3.2.5.2.2

Multiply $\tan (θ)$ by $1$ .

$\frac{y'}{y} = \tan (θ) + \sec^{2} (θ) \cot (θ)$

Step 3.2.5.2.3

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

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \sec^{2} (θ) \cot (θ)$

Step 3.2.5.2.4

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

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + {(\frac{1}{\cos (θ)})}^{2} \cot (θ)$

Step 3.2.5.2.5

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

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \frac{1^{2}}{\cos^{2} (θ)} \cot (θ)$

Step 3.2.5.2.6

One to any power is one.

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \frac{1}{\cos^{2} (θ)} \cot (θ)$

Step 3.2.5.2.7

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

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \frac{1}{\cos^{2} (θ)} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 3.2.5.2.8

Combine.

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \frac{1 \cos (θ)}{\cos^{2} (θ) \sin (θ)}$

Step 3.2.5.2.9

Cancel the common factor of $\cos (θ)$ and $\cos^{2} (θ)$ .

Tap for more steps...

Step 3.2.5.2.9.1

Factor $\cos (θ)$ out of $1 \cos (θ)$ .

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \frac{\cos (θ) \cdot 1}{\cos^{2} (θ) \sin (θ)}$

Step 3.2.5.2.9.2

Cancel the common factors.

Tap for more steps...

Step 3.2.5.2.9.2.1

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

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

Step 3.2.5.2.9.2.2

Cancel the common factor.

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

Step 3.2.5.2.9.2.3

Rewrite the expression.

$\frac{y'}{y} = \frac{\sin (θ)}{\cos (θ)} + \frac{1}{\cos (θ) \sin (θ)}$

Step 3.2.5.3

Simplify each term.

Tap for more steps...

Step 3.2.5.3.1

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

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

Step 3.2.5.3.2

Separate fractions.

$\frac{y'}{y} = \tan (θ) + \frac{1}{\sin (θ)} \cdot \frac{1}{\cos (θ)}$

Step 3.2.5.3.3

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

$\frac{y'}{y} = \tan (θ) + \frac{1}{\sin (θ)} \sec (θ)$

Step 3.2.5.3.4

Convert from $\frac{1}{\sin (θ)}$ to $\csc (θ)$ .

$\frac{y'}{y} = \tan (θ) + \csc (θ) \sec (θ)$

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = (\tan (θ) + \csc (θ) \sec (θ)) (\sec (θ) \tan (θ))$

Step 5

Simplify the right hand side.

Tap for more steps...

Step 5.1

Apply the distributive property.

$y' = \tan (θ) (\sec (θ) \tan (θ)) + \csc (θ) \sec (θ) (\sec (θ) \tan (θ))$

Step 5.2

Multiply $\tan (θ) (\sec (θ) \tan (θ))$ .

Tap for more steps...

Step 5.2.1

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

$y' = \tan^{1} (θ) \tan (θ) \sec (θ) + \csc (θ) \sec (θ) (\sec (θ) \tan (θ))$

Step 5.2.2

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

$y' = \tan^{1} (θ) \tan^{1} (θ) \sec (θ) + \csc (θ) \sec (θ) (\sec (θ) \tan (θ))$

Step 5.2.3

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

$y' = {\tan (θ)}^{1 + 1} \sec (θ) + \csc (θ) \sec (θ) (\sec (θ) \tan (θ))$

Step 5.2.4

Add $1$ and $1$ .

$y' = \tan^{2} (θ) \sec (θ) + \csc (θ) \sec (θ) (\sec (θ) \tan (θ))$

Step 5.3

Multiply $\csc (θ) \sec (θ) (\sec (θ) \tan (θ))$ .

Tap for more steps...

Step 5.3.1

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

$y' = \tan^{2} (θ) \sec (θ) + \csc (θ) (\sec^{1} (θ) \sec (θ)) \tan (θ)$

Step 5.3.2

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

$y' = \tan^{2} (θ) \sec (θ) + \csc (θ) (\sec^{1} (θ) \sec^{1} (θ)) \tan (θ)$

Step 5.3.3

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

$y' = \tan^{2} (θ) \sec (θ) + \csc (θ) {\sec (θ)}^{1 + 1} \tan (θ)$

Step 5.3.4

Add $1$ and $1$ .

$y' = \tan^{2} (θ) \sec (θ) + \csc (θ) \sec^{2} (θ) \tan (θ)$