Calculus Examples

$F (x) = 5 x^{- 2} \cdot e^{x}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{- 2} e^{x}$ with respect to $x$ is $5 \frac{d}{d x} [x^{- 2} e^{x}]$ .

$5 \frac{d}{d x} [x^{- 2} e^{x}]$

Step 2

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 $f (x) = x^{- 2}$ and $g (x) = e^{x}$ .

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

Step 3

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

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

Step 4

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

$5 (x^{- 2} e^{x} + e^{x} (- 2 x^{- 3}))$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (\frac{1}{x^{2}} e^{x} + e^{x} (- 2 x^{- 3}))$

Step 5.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (\frac{1}{x^{2}} e^{x} + e^{x} (- 2 \frac{1}{x^{3}}))$

Step 5.3

Apply the distributive property.

$5 (\frac{1}{x^{2}} e^{x}) + 5 (e^{x} (- 2 \frac{1}{x^{3}}))$

Step 5.4

Combine terms.

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Step 5.4.1

Combine $\frac{1}{x^{2}}$ and $e^{x}$ .

$5 \frac{e^{x}}{x^{2}} + 5 (e^{x} (- 2 \frac{1}{x^{3}}))$

Step 5.4.2

Combine $5$ and $\frac{e^{x}}{x^{2}}$ .

$\frac{5 e^{x}}{x^{2}} + 5 (e^{x} (- 2 \frac{1}{x^{3}}))$

Step 5.4.3

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$\frac{5 e^{x}}{x^{2}} + 5 (e^{x} \frac{- 2}{x^{3}})$

Step 5.4.4

Move the negative in front of the fraction.

$\frac{5 e^{x}}{x^{2}} + 5 (e^{x} (- \frac{2}{x^{3}}))$

Step 5.4.5

Combine $e^{x}$ and $\frac{2}{x^{3}}$ .

$\frac{5 e^{x}}{x^{2}} + 5 (- \frac{e^{x} \cdot 2}{x^{3}})$

Step 5.4.6

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

$\frac{5 e^{x}}{x^{2}} + 5 (- \frac{2 \cdot e^{x}}{x^{3}})$

Step 5.4.7

Multiply $- 1$ by $5$ .

$\frac{5 e^{x}}{x^{2}} - 5 \frac{2 e^{x}}{x^{3}}$

Step 5.4.8

Combine $- 5$ and $\frac{2 e^{x}}{x^{3}}$ .

$\frac{5 e^{x}}{x^{2}} + \frac{- 5 (2 e^{x})}{x^{3}}$

Step 5.4.9

Multiply $2$ by $- 5$ .

$\frac{5 e^{x}}{x^{2}} + \frac{- 10 e^{x}}{x^{3}}$

Step 5.4.10

Move the negative in front of the fraction.

$\frac{5 e^{x}}{x^{2}} - \frac{10 e^{x}}{x^{3}}$