Calculus Examples

$f (x) = (\frac{e^{2 x}}{x^{6}})$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{2 x}$ and $g (x) = x^{6}$ .

$\frac{x^{6} \frac{d}{d x} [e^{2 x}] - e^{2 x} \frac{d}{d x} [x^{6}]}{{(x^{6})}^{2}}$

Step 2

Multiply the exponents in ${(x^{6})}^{2}$ .

Tap for more steps...

Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{6} \frac{d}{d x} [e^{2 x}] - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{6 \cdot 2}}$

Step 2.2

Multiply $6$ by $2$ .

$\frac{x^{6} \frac{d}{d x} [e^{2 x}] - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 3

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 $g (x) = 2 x$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{x^{6} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 3.2

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

$\frac{x^{6} (e^{u} \frac{d}{d x} [2 x]) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 3.3

Replace all occurrences of $u$ with $2 x$ .

$\frac{x^{6} (e^{2 x} \frac{d}{d x} [2 x]) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

$\frac{x^{6} (e^{2 x} (2 \frac{d}{d x} [x])) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2

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

$\frac{x^{6} (e^{2 x} (2 \cdot 1)) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.3

Simplify the expression.

Tap for more steps...

Step 4.3.1

Multiply $2$ by $1$ .

$\frac{x^{6} (e^{2 x} \cdot 2) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.3.2

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

$\frac{x^{6} (2 \cdot e^{2 x}) - e^{2 x} \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{x^{6} (2 e^{2 x}) - e^{2 x} (6 x^{5})}{x^{12}}$

Step 4.5

Multiply $6$ by $- 1$ .

$\frac{x^{6} (2 e^{2 x}) - 6 e^{2 x} x^{5}}{x^{12}}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{6} e^{2 x} - 6 e^{2 x} x^{5}}{x^{12}}$

Step 5.1.2

Reorder factors in $2 x^{6} e^{2 x} - 6 e^{2 x} x^{5}$ .

$\frac{2 x^{6} e^{2 x} - 6 x^{5} e^{2 x}}{x^{12}}$

Step 5.2

Reorder terms.

$\frac{2 e^{2 x} x^{6} - 6 e^{2 x} x^{5}}{x^{12}}$

Step 5.3

Factor $2 e^{2 x} x^{5}$ out of $2 e^{2 x} x^{6} - 6 e^{2 x} x^{5}$ .

Tap for more steps...

Step 5.3.1

Factor $2 e^{2 x} x^{5}$ out of $2 e^{2 x} x^{6}$ .

$\frac{2 e^{2 x} x^{5} (x) - 6 e^{2 x} x^{5}}{x^{12}}$

Step 5.3.2

Factor $2 e^{2 x} x^{5}$ out of $- 6 e^{2 x} x^{5}$ .

$\frac{2 e^{2 x} x^{5} (x) + 2 e^{2 x} x^{5} (- 3)}{x^{12}}$

Step 5.3.3

Factor $2 e^{2 x} x^{5}$ out of $2 e^{2 x} x^{5} (x) + 2 e^{2 x} x^{5} (- 3)$ .

$\frac{2 e^{2 x} x^{5} (x - 3)}{x^{12}}$

Step 5.4

Cancel the common factor of $x^{5}$ and $x^{12}$ .

Tap for more steps...

Step 5.4.1

Factor $x^{5}$ out of $2 e^{2 x} x^{5} (x - 3)$ .

$\frac{x^{5} (2 e^{2 x} (x - 3))}{x^{12}}$

Step 5.4.2

Cancel the common factors.

Tap for more steps...

Step 5.4.2.1

Factor $x^{5}$ out of $x^{12}$ .

$\frac{x^{5} (2 e^{2 x} (x - 3))}{x^{5} x^{7}}$

Step 5.4.2.2

Cancel the common factor.

$\frac{x^{5} (2 e^{2 x} (x - 3))}{x^{5} x^{7}}$

Step 5.4.2.3

Rewrite the expression.

$\frac{2 e^{2 x} (x - 3)}{x^{7}}$