Calculus Examples

$a e^{- a x}$

Step 1

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

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

Step 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 $g (x) = - a x$ .

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Step 2.1

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

$a (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- a x])$

Step 2.2

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

$a (e^{u} \frac{d}{d x} [- a x])$

Step 2.3

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

$a (e^{- a x} \frac{d}{d x} [- a x])$

Step 3

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

$a (e^{- a x} (- a \frac{d}{d x} [x]))$

Step 4

Raise $a$ to the power of $1$ .

$a^{1} a (e^{- a x} (- \frac{d}{d x} [x]))$

Step 5

Raise $a$ to the power of $1$ .

$a^{1} a^{1} (e^{- a x} (- \frac{d}{d x} [x]))$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Simplify the expression.

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Step 9.1

Multiply $- 1$ by $1$ .

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

Step 9.2

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

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

Step 9.3

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

$a^{2} (- e^{- a x})$

Step 10

Simplify.

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Step 10.1

Reorder the factors of $a^{2} (- e^{- a x})$ .

$- e^{- a x} a^{2}$

Step 10.2

Reorder factors in $- e^{- a x} a^{2}$ .

$- a^{2} e^{- a x}$