Calculus Examples

$7 \sec (x) + 3 e^{x} \tan (x)$

Step 1

By the Sum Rule, the derivative of $7 \sec (x) + 3 e^{x} \tan (x)$ with respect to $x$ is $\frac{d}{d x} [7 \sec (x)] + \frac{d}{d x} [3 e^{x} \tan (x)]$ .

$\frac{d}{d x} [7 \sec (x)] + \frac{d}{d x} [3 e^{x} \tan (x)]$

Step 2

Evaluate $\frac{d}{d x} [7 \sec (x)]$ .

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \sec (x)$ with respect to $x$ is $7 \frac{d}{d x} [\sec (x)]$ .

$7 \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [3 e^{x} \tan (x)]$

Step 2.2

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

$7 \sec (x) \tan (x) + \frac{d}{d x} [3 e^{x} \tan (x)]$

Step 3

Evaluate $\frac{d}{d x} [3 e^{x} \tan (x)]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x} \tan (x)$ with respect to $x$ is $3 \frac{d}{d x} [e^{x} \tan (x)]$ .

$7 \sec (x) \tan (x) + 3 \frac{d}{d x} [e^{x} \tan (x)]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \tan (x)$ .

$7 \sec (x) \tan (x) + 3 (e^{x} \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [e^{x}])$

Step 3.3

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

$7 \sec (x) \tan (x) + 3 (e^{x} \sec^{2} (x) + \tan (x) \frac{d}{d x} [e^{x}])$

Step 3.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$7 \sec (x) \tan (x) + 3 (e^{x} \sec^{2} (x) + \tan (x) e^{x})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$7 \sec (x) \tan (x) + 3 (e^{x} \sec^{2} (x)) + 3 (\tan (x) e^{x})$

Step 4.2

Remove unnecessary parentheses.

$7 \sec (x) \tan (x) + 3 e^{x} \sec^{2} (x) + 3 \tan (x) e^{x}$

Step 4.3

Reorder terms.

$7 \sec (x) \tan (x) + 3 e^{x} \sec^{2} (x) + 3 e^{x} \tan (x)$