Calculus Examples

$y = {(x^{6} \ln (x))}^{5}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{6} \ln (x)$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $x^{6} \ln (x)$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{6} \ln (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [x^{6} \ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $x^{6} \ln (x)$ .

$5 {(x^{6} \ln (x))}^{4} \frac{d}{d x} [x^{6} \ln (x)]$

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 $f (x) = x^{6}$ and $g (x) = \ln (x)$ .

$5 {(x^{6} \ln (x))}^{4} (x^{6} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{6}])$

Step 3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$5 {(x^{6} \ln (x))}^{4} (x^{6} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4

Differentiate using the Power Rule.

Tap for more steps...

Step 4.1

Combine $x^{6}$ and $\frac{1}{x}$ .

$5 {(x^{6} \ln (x))}^{4} (\frac{x^{6}}{x} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.2

Cancel the common factor of $x^{6}$ and $x$ .

Tap for more steps...

Step 4.2.1

Factor $x$ out of $x^{6}$ .

$5 {(x^{6} \ln (x))}^{4} (\frac{x \cdot x^{5}}{x} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

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

$5 {(x^{6} \ln (x))}^{4} (\frac{x \cdot x^{5}}{x^{1}} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$5 {(x^{6} \ln (x))}^{4} (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.2.2.3

Cancel the common factor.

$5 {(x^{6} \ln (x))}^{4} (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.2.2.4

Rewrite the expression.

$5 {(x^{6} \ln (x))}^{4} (\frac{x^{5}}{1} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.2.2.5

Divide $x^{5}$ by $1$ .

$5 {(x^{6} \ln (x))}^{4} (x^{5} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 4.3

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

$5 {(x^{6} \ln (x))}^{4} (x^{5} + \ln (x) (6 x^{5}))$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the product rule to $x^{6} \ln (x)$ .

$5 ({(x^{6})}^{4} \ln^{4} (x)) (x^{5} + \ln (x) (6 x^{5}))$

Step 5.2

Apply the distributive property.

$5 ({(x^{6})}^{4} \ln^{4} (x)) x^{5} + 5 ({(x^{6})}^{4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3

Combine terms.

Tap for more steps...

Step 5.3.1

Multiply the exponents in ${(x^{6})}^{4}$ .

Tap for more steps...

Step 5.3.1.1

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

$5 (x^{6 \cdot 4} \ln^{4} (x)) x^{5} + 5 ({(x^{6})}^{4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.1.2

Multiply $6$ by $4$ .

$5 (x^{24} \ln^{4} (x)) x^{5} + 5 ({(x^{6})}^{4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.2

Multiply $x^{24}$ by $x^{5}$ by adding the exponents.

Tap for more steps...

Step 5.3.2.1

Move $x^{5}$ .

$5 (x^{5} x^{24}) \ln^{4} (x) + 5 ({(x^{6})}^{4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.2.2

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

$5 x^{5 + 24} \ln^{4} (x) + 5 ({(x^{6})}^{4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.2.3

Add $5$ and $24$ .

$5 x^{29} \ln^{4} (x) + 5 ({(x^{6})}^{4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.3

Multiply the exponents in ${(x^{6})}^{4}$ .

Tap for more steps...

Step 5.3.3.1

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

$5 x^{29} \ln^{4} (x) + 5 (x^{6 \cdot 4} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.3.2

Multiply $6$ by $4$ .

$5 x^{29} \ln^{4} (x) + 5 (x^{24} \ln^{4} (x)) (\ln (x) (6 x^{5}))$

Step 5.3.4

Multiply $6$ by $5$ .

$5 x^{29} \ln^{4} (x) + 30 x^{24} \ln^{4} (x) (\ln (x) (x^{5}))$

Step 5.3.5

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

$5 x^{29} \ln^{4} (x) + 30 x^{5 + 24} \ln^{4} (x) \ln (x)$

Step 5.3.6

Add $5$ and $24$ .

$5 x^{29} \ln^{4} (x) + 30 x^{29} \ln^{4} (x) \ln (x)$

Step 5.3.7

Raise $\ln (x)$ to the power of $1$ .

$5 x^{29} \ln^{4} (x) + 30 x^{29} (\ln^{1} (x) \ln^{4} (x))$

Step 5.3.8

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

$5 x^{29} \ln^{4} (x) + 30 x^{29} {\ln (x)}^{1 + 4}$

Step 5.3.9

Add $1$ and $4$ .

$5 x^{29} \ln^{4} (x) + 30 x^{29} \ln^{5} (x)$