Calculus Examples

$y = \sec (θ) \tan (θ)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d θ} (y) = \frac{d}{d θ} (\sec (θ) \tan (θ))$

Step 2

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

$y'$

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

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

Step 3.2

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

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

Step 3.3

Multiply $\sec (θ)$ by $\sec^{2} (θ)$ by adding the exponents.

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Step 3.3.1

Multiply $\sec (θ)$ by $\sec^{2} (θ)$ .

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Step 3.3.1.1

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

$\sec^{1} (θ) \sec^{2} (θ) + \tan (θ) \frac{d}{d θ} [\sec (θ)]$

Step 3.3.1.2

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

${\sec (θ)}^{1 + 2} + \tan (θ) \frac{d}{d θ} [\sec (θ)]$

Step 3.3.2

Add $1$ and $2$ .

$\sec^{3} (θ) + \tan (θ) \frac{d}{d θ} [\sec (θ)]$

Step 3.4

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

$\sec^{3} (θ) + \tan (θ) (\sec (θ) \tan (θ))$

Step 3.5

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

$\sec^{3} (θ) + \tan^{1} (θ) \tan (θ) \sec (θ)$

Step 3.6

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

$\sec^{3} (θ) + \tan^{1} (θ) \tan^{1} (θ) \sec (θ)$

Step 3.7

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

$\sec^{3} (θ) + {\tan (θ)}^{1 + 1} \sec (θ)$

Step 3.8

Add $1$ and $1$ .

$\sec^{3} (θ) + \tan^{2} (θ) \sec (θ)$

Step 3.9

Reorder terms.

$\tan^{2} (θ) \sec (θ) + \sec^{3} (θ)$

Step 4

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

$y' = \tan^{2} (θ) \sec (θ) + \sec^{3} (θ)$

Step 5

Replace $y'$ with $\frac{d y}{d θ}$ .

$\frac{d y}{d θ} = \tan^{2} (θ) \sec (θ) + \sec^{3} (θ)$