Calculus Examples

$y = 2 e^{x} - x + 5 \sin (x) - 12$

Step 1

By the Sum Rule, the derivative of $2 e^{x} - x + 5 \sin (x) - 12$ with respect to $x$ is $\frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$ .

$\frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$

Step 2

Evaluate $\frac{d}{d x} [2 e^{x}]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{x}]$ .

$2 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$

Step 2.2

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

$2 e^{x} + \frac{d}{d x} [- x] + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$

Step 3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$2 e^{x} - \frac{d}{d x} [x] + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 e^{x} - 1 \cdot 1 + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$

Step 3.3

Multiply $- 1$ by $1$ .

$2 e^{x} - 1 + \frac{d}{d x} [5 \sin (x)] + \frac{d}{d x} [- 12]$

Step 4

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

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Step 4.1

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

$2 e^{x} - 1 + 5 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- 12]$

Step 4.2

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

$2 e^{x} - 1 + 5 \cos (x) + \frac{d}{d x} [- 12]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$2 e^{x} - 1 + 5 \cos (x) + 0$

Step 5.2

Add $2 e^{x} - 1 + 5 \cos (x)$ and $0$ .

$2 e^{x} - 1 + 5 \cos (x)$