Calculus Examples

$(1 - 4 e^{x}) x^{2}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 1 - 4 e^{x}$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Move $2$ to the left of $1 - 4 e^{x}$ .

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

Step 2.3

By the Sum Rule, the derivative of $1 - 4 e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 e^{x}]$ .

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

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 (1 - 4 e^{x}) x + x^{2} (0 + \frac{d}{d x} [- 4 e^{x}])$

Step 2.5

Add $0$ and $\frac{d}{d x} [- 4 e^{x}]$ .

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

Step 2.6

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $2$ by $1$ .

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

Step 4.3.2

Multiply $- 4$ by $2$ .

$2 x - 8 e^{x} x + x^{2} (- 4 e^{x})$

Step 4.4

Reorder terms.

$- 8 e^{x} x - 4 e^{x} x^{2} + 2 x$

Step 4.5

Reorder factors in $- 8 e^{x} x - 4 e^{x} x^{2} + 2 x$ .

$- 8 x e^{x} - 4 x^{2} e^{x} + 2 x$