Calculus Examples

$x e^{a x}$

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = e^{a}$ and $g (a) = a x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Multiply $e^{a x}$ by $1$ .

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