Calculus Examples

$f (x) = 3 \sin (x) - 4 \cos (x) + 3 e^{x} - 5 \ln (x)$

Step 1

By the Sum Rule, the derivative of $3 \sin (x) - 4 \cos (x) + 3 e^{x} - 5 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [3 \sin (x)] + \frac{d}{d x} [- 4 \cos (x)] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$ .

$\frac{d}{d x} [3 \sin (x)] + \frac{d}{d x} [- 4 \cos (x)] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 2

Evaluate $\frac{d}{d x} [3 \sin (x)]$ .

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Step 2.1

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

$3 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- 4 \cos (x)] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 2.2

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

$3 \cos (x) + \frac{d}{d x} [- 4 \cos (x)] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 3

Evaluate $\frac{d}{d x} [- 4 \cos (x)]$ .

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Step 3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \cos (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\cos (x)]$ .

$3 \cos (x) - 4 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 3.2

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

$3 \cos (x) - 4 (- \sin (x)) + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 3.3

Multiply $- 1$ by $- 4$ .

$3 \cos (x) + 4 \sin (x) + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 4

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

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Step 4.1

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

$3 \cos (x) + 4 \sin (x) + 3 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 5 \ln (x)]$

Step 4.2

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

$3 \cos (x) + 4 \sin (x) + 3 e^{x} + \frac{d}{d x} [- 5 \ln (x)]$

Step 5

Evaluate $\frac{d}{d x} [- 5 \ln (x)]$ .

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Step 5.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \ln (x)$ with respect to $x$ is $- 5 \frac{d}{d x} [\ln (x)]$ .

$3 \cos (x) + 4 \sin (x) + 3 e^{x} - 5 \frac{d}{d x} [\ln (x)]$

Step 5.2

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

$3 \cos (x) + 4 \sin (x) + 3 e^{x} - 5 \frac{1}{x}$

Step 5.3

Combine $- 5$ and $\frac{1}{x}$ .

$3 \cos (x) + 4 \sin (x) + 3 e^{x} + \frac{- 5}{x}$

Step 5.4

Move the negative in front of the fraction.

$3 \cos (x) + 4 \sin (x) + 3 e^{x} - \frac{5}{x}$

Step 6

Reorder terms.

$- \frac{5}{x} + 3 e^{x} + 3 \cos (x) + 4 \sin (x)$