Calculus Examples

$y = 7 x^{2} + 2 x - e^{4 x}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $7 x^{2} + 2 x - e^{4 x}$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- e^{4 x}]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- e^{4 x}]$

Step 1.2

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 1.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- e^{4 x}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$7 (2 x) + \frac{d}{d x} [2 x] + \frac{d}{d x} [- e^{4 x}]$

Step 1.2.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- e^{4 x}]$

Step 1.3

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

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Step 1.3.1

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

$14 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- e^{4 x}]$

Step 1.3.2

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

$14 x + 2 \cdot 1 + \frac{d}{d x} [- e^{4 x}]$

Step 1.3.3

Multiply $2$ by $1$ .

$14 x + 2 + \frac{d}{d x} [- e^{4 x}]$

Step 1.4

Evaluate $\frac{d}{d x} [- e^{4 x}]$ .

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Step 1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{4 x}$ with respect to $x$ is $- \frac{d}{d x} [e^{4 x}]$ .

$14 x + 2 - \frac{d}{d x} [e^{4 x}]$

Step 1.4.2

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 $g (x) = 4 x$ .

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Step 1.4.2.1

To apply the Chain Rule, set $u$ as $4 x$ .

$14 x + 2 - (\frac{d}{d u} [e^{u}] \frac{d}{d x} [4 x])$

Step 1.4.2.2

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

$14 x + 2 - (e^{u} \frac{d}{d x} [4 x])$

Step 1.4.2.3

Replace all occurrences of $u$ with $4 x$ .

$14 x + 2 - (e^{4 x} \frac{d}{d x} [4 x])$

Step 1.4.3

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

$14 x + 2 - (e^{4 x} (4 \frac{d}{d x} [x]))$

Step 1.4.4

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

$14 x + 2 - (e^{4 x} (4 \cdot 1))$

Step 1.4.5

Multiply $4$ by $1$ .

$14 x + 2 - (e^{4 x} \cdot 4)$

Step 1.4.6

Move $4$ to the left of $e^{4 x}$ .

$14 x + 2 - (4 \cdot e^{4 x})$

Step 1.4.7

Multiply $4$ by $- 1$ .

$f' (x) = 14 x + 2 - 4 e^{4 x}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $14 x + 2 - 4 e^{4 x}$ with respect to $x$ is $\frac{d}{d x} [14 x] + \frac{d}{d x} [2] + \frac{d}{d x} [- 4 e^{4 x}]$ .

$\frac{d}{d x} [14 x] + \frac{d}{d x} [2] + \frac{d}{d x} [- 4 e^{4 x}]$

Step 2.2

Evaluate $\frac{d}{d x} [14 x]$ .

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Step 2.2.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$14 \frac{d}{d x} [x] + \frac{d}{d x} [2] + \frac{d}{d x} [- 4 e^{4 x}]$

Step 2.2.2

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

$14 \cdot 1 + \frac{d}{d x} [2] + \frac{d}{d x} [- 4 e^{4 x}]$

Step 2.2.3

Multiply $14$ by $1$ .

$14 + \frac{d}{d x} [2] + \frac{d}{d x} [- 4 e^{4 x}]$

Step 2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$14 + 0 + \frac{d}{d x} [- 4 e^{4 x}]$

Step 2.4

Evaluate $\frac{d}{d x} [- 4 e^{4 x}]$ .

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Step 2.4.1

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

$14 + 0 - 4 \frac{d}{d x} [e^{4 x}]$

Step 2.4.2

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 $g (x) = 4 x$ .

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Step 2.4.2.1

To apply the Chain Rule, set $u$ as $4 x$ .

$14 + 0 - 4 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [4 x])$

Step 2.4.2.2

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

$14 + 0 - 4 (e^{u} \frac{d}{d x} [4 x])$

Step 2.4.2.3

Replace all occurrences of $u$ with $4 x$ .

$14 + 0 - 4 (e^{4 x} \frac{d}{d x} [4 x])$

Step 2.4.3

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

$14 + 0 - 4 (e^{4 x} (4 \frac{d}{d x} [x]))$

Step 2.4.4

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

$14 + 0 - 4 (e^{4 x} (4 \cdot 1))$

Step 2.4.5

Multiply $4$ by $1$ .

$14 + 0 - 4 (e^{4 x} \cdot 4)$

Step 2.4.6

Move $4$ to the left of $e^{4 x}$ .

$14 + 0 - 4 (4 \cdot e^{4 x})$

Step 2.4.7

Multiply $4$ by $- 4$ .

$14 + 0 - 16 e^{4 x}$

Step 2.5

Add $14$ and $0$ .

$f'' (x) = 14 - 16 e^{4 x}$