Calculus Examples

$y = e^{3 x} + 3 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{3 x} + 3 x)$

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 $e^{3 x} + 3 x$ with respect to $x$ is $\frac{d}{d x} [e^{3 x}] + \frac{d}{d x} [3 x]$ .

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

Step 3.2

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

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Step 3.2.1

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 $g (x) = 3 x$ .

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Step 3.2.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x] + \frac{d}{d x} [3 x]$

Step 3.2.1.2

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

$e^{u} \frac{d}{d x} [3 x] + \frac{d}{d x} [3 x]$

Step 3.2.1.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $3$ by $1$ .

$e^{3 x} \cdot 3 + \frac{d}{d x} [3 x]$

Step 3.2.5

Move $3$ to the left of $e^{3 x}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

$3 e^{3 x} + 3 \cdot 1$

Step 3.3.3

Multiply $3$ by $1$ .

$3 e^{3 x} + 3$

Step 4

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

$y' = 3 e^{3 x} + 3$

Step 5

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

$\frac{d y}{d x} = 3 e^{3 x} + 3$