Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(5 - x)}^{6}$ .

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

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [{(5 - x)}^{6}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(5 - x)}^{6}$ .

$\frac{1}{{(5 - x)}^{6}} \frac{d}{d x} [{(5 - x)}^{6}]$

Step 2

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

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $5 - x$ .

$\frac{1}{{(5 - x)}^{6}} (\frac{d}{d u_{2}} [{u_{2}}^{6}] \frac{d}{d x} [5 - x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 6$ .

$\frac{1}{{(5 - x)}^{6}} (6 {u_{2}}^{5} \frac{d}{d x} [5 - x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $5 - x$ .

$\frac{1}{{(5 - x)}^{6}} (6 {(5 - x)}^{5} \frac{d}{d x} [5 - x])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Combine $6$ and $\frac{1}{{(5 - x)}^{6}}$ .

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

Step 3.2

Simplify terms.

Tap for more steps...

Step 3.2.1

Combine ${(5 - x)}^{5}$ and $\frac{6}{{(5 - x)}^{6}}$ .

$\frac{{(5 - x)}^{5} \cdot 6}{{(5 - x)}^{6}} \frac{d}{d x} [5 - x]$

Step 3.2.2

Move $6$ to the left of ${(5 - x)}^{5}$ .

$\frac{6 \cdot {(5 - x)}^{5}}{{(5 - x)}^{6}} \frac{d}{d x} [5 - x]$

Step 3.2.3

Cancel the common factor of ${(5 - x)}^{5}$ and ${(5 - x)}^{6}$ .

Tap for more steps...

Step 3.2.3.1

Factor ${(5 - x)}^{5}$ out of $6 {(5 - x)}^{5}$ .

$\frac{{(5 - x)}^{5} \cdot 6}{{(5 - x)}^{6}} \frac{d}{d x} [5 - x]$

Step 3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.2.1

Factor ${(5 - x)}^{5}$ out of ${(5 - x)}^{6}$ .

$\frac{{(5 - x)}^{5} \cdot 6}{{(5 - x)}^{5} (5 - x)} \frac{d}{d x} [5 - x]$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{{(5 - x)}^{5} \cdot 6}{{(5 - x)}^{5} (5 - x)} \frac{d}{d x} [5 - x]$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{6}{5 - x} \frac{d}{d x} [5 - x]$

Step 3.3

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$\frac{6}{5 - x} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 3.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{6}{5 - x} (0 + \frac{d}{d x} [- x])$

Step 3.5

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{6}{5 - x} \frac{d}{d x} [- x]$

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{6}{5 - x} (- \frac{d}{d x} [x])$

Step 3.7

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

$\frac{6}{5 - x} (- 1 \cdot 1)$

Step 3.8

Combine fractions.

Tap for more steps...

Step 3.8.1

Multiply $- 1$ by $1$ .

$\frac{6}{5 - x} \cdot - 1$

Step 3.8.2

Combine $\frac{6}{5 - x}$ and $- 1$ .

$\frac{6 \cdot - 1}{5 - x}$

Step 3.8.3

Simplify the expression.

Tap for more steps...

Step 3.8.3.1

Multiply $6$ by $- 1$ .

$\frac{- 6}{5 - x}$

Step 3.8.3.2

Move the negative in front of the fraction.

$- \frac{6}{5 - x}$