Calculus Examples

$\sin (x) + e^{- x} + 3 x$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - x$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $- x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $- x$ .

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

Step 3.2

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

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

Step 3.3

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

$\cos (x) + e^{- x} (- 1 \cdot 1) + \frac{d}{d x} [3 x]$

Step 3.4

Multiply $- 1$ by $1$ .

$\cos (x) + e^{- x} \cdot - 1 + \frac{d}{d x} [3 x]$

Step 3.5

Move $- 1$ to the left of $e^{- x}$ .

$\cos (x) - 1 \cdot e^{- x} + \frac{d}{d x} [3 x]$

Step 3.6

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$\cos (x) - e^{- x} + 3 \cdot 1$

Step 4.3

Multiply $3$ by $1$ .

$\cos (x) - e^{- x} + 3$

Step 5

Reorder terms.

$3 - e^{- x} + \cos (x)$