Calculus Examples

$y = \tan (x) + \sec (x)$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$\sec^{2} (x) + \sec (x) \tan (x)$

Step 3.4

Reorder terms.

$\sec (x) \tan (x) + \sec^{2} (x)$

Step 4

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

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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