Calculus Examples

$y = \log (| 3 - 5 x |)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = | 3 - 5 x |$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $| 3 - 5 x |$ .

$\frac{d}{d u_{1}} [\log (u_{1})] \frac{d}{d x} [| 3 - 5 x |]$

Step 1.2

The derivative of $\log (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (10)}$ .

$\frac{1}{u_{1} \ln (10)} \frac{d}{d x} [| 3 - 5 x |]$

Step 1.3

Replace all occurrences of $u_{1}$ with $| 3 - 5 x |$ .

$\frac{1}{| 3 - 5 x | \ln (10)} \frac{d}{d x} [| 3 - 5 x |]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 3 - 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $3 - 5 x$ .

$\frac{1}{| 3 - 5 x | \ln (10)} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [3 - 5 x])$

Step 2.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{| 3 - 5 x | \ln (10)} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [3 - 5 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $3 - 5 x$ .

$\frac{1}{| 3 - 5 x | \ln (10)} (\frac{3 - 5 x}{| 3 - 5 x |} \frac{d}{d x} [3 - 5 x])$

Step 3

Multiply $\frac{3 - 5 x}{| 3 - 5 x |}$ by $\frac{1}{| 3 - 5 x | \ln (10)}$ .

$\frac{3 - 5 x}{| 3 - 5 x | (| 3 - 5 x | \ln (10))} \frac{d}{d x} [3 - 5 x]$

Step 4

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{3 - 5 x}{| (3 - 5 x) (3 - 5 x) | \ln (10)} \frac{d}{d x} [3 - 5 x]$

Step 5

Raise $3 - 5 x$ to the power of $1$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{1} (3 - 5 x) | \ln (10)} \frac{d}{d x} [3 - 5 x]$

Step 6

Raise $3 - 5 x$ to the power of $1$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{1} {(3 - 5 x)}^{1} | \ln (10)} \frac{d}{d x} [3 - 5 x]$

Step 7

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

$\frac{3 - 5 x}{| {(3 - 5 x)}^{1 + 1} | \ln (10)} \frac{d}{d x} [3 - 5 x]$

Step 8

Add $1$ and $1$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{2} | \ln (10)} \frac{d}{d x} [3 - 5 x]$

Step 9

By the Sum Rule, the derivative of $3 - 5 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x]$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{2} | \ln (10)} (\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x])$

Step 10

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{2} | \ln (10)} (0 + \frac{d}{d x} [- 5 x])$

Step 11

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{2} | \ln (10)} \frac{d}{d x} [- 5 x]$

Step 12

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$\frac{3 - 5 x}{| {(3 - 5 x)}^{2} | \ln (10)} (- 5 \frac{d}{d x} [x])$

Step 13

Combine fractions.

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Step 13.1

Combine $- 5$ and $\frac{3 - 5 x}{| {(3 - 5 x)}^{2} | \ln (10)}$ .

$\frac{- 5 (3 - 5 x)}{| {(3 - 5 x)}^{2} | \ln (10)} \frac{d}{d x} [x]$

Step 13.2

Move the negative in front of the fraction.

$- \frac{5 (3 - 5 x)}{| {(3 - 5 x)}^{2} | \ln (10)} \frac{d}{d x} [x]$

Step 14

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

$- \frac{5 (3 - 5 x)}{| {(3 - 5 x)}^{2} | \ln (10)} \cdot 1$

Step 15

Multiply $- 1$ by $1$ .

$- \frac{5 (3 - 5 x)}{| {(3 - 5 x)}^{2} | \ln (10)}$