Calculus Examples

$z = \frac{\ln (3.4 x^{4} + 7.1)}{y + 3.7}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (z) = \frac{d}{d x} (\frac{\ln (3.4 x^{4} + 7.1)}{y + 3.7})$

Step 2

The derivative of $z$ with respect to $x$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $\frac{1}{y + 3.7}$ is constant with respect to $x$ , the derivative of $\frac{\ln (3.4 x^{4} + 7.1)}{y + 3.7}$ with respect to $x$ is $\frac{1}{y + 3.7} \frac{d}{d x} [\ln (3.4 x^{4} + 7.1)]$ .

$\frac{1}{y + 3.7} \frac{d}{d x} [\ln (3.4 x^{4} + 7.1)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3.4 x^{4} + 7.1$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3.4 x^{4} + 7.1$ .

$\frac{1}{y + 3.7} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3.4 x^{4} + 7.1])$

Step 3.2.2

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

$\frac{1}{y + 3.7} (\frac{1}{u} \frac{d}{d x} [3.4 x^{4} + 7.1])$

Step 3.2.3

Replace all occurrences of $u$ with $3.4 x^{4} + 7.1$ .

$\frac{1}{y + 3.7} (\frac{1}{3.4 x^{4} + 7.1} \frac{d}{d x} [3.4 x^{4} + 7.1])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $\frac{1}{3.4 x^{4} + 7.1}$ by $\frac{1}{y + 3.7}$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} \frac{d}{d x} [3.4 x^{4} + 7.1]$

Step 3.3.2

By the Sum Rule, the derivative of $3.4 x^{4} + 7.1$ with respect to $x$ is $\frac{d}{d x} [3.4 x^{4}] + \frac{d}{d x} [7.1]$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} (\frac{d}{d x} [3.4 x^{4}] + \frac{d}{d x} [7.1])$

Step 3.3.3

Since $3.4$ is constant with respect to $x$ , the derivative of $3.4 x^{4}$ with respect to $x$ is $3.4 \frac{d}{d x} [x^{4}]$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} (3.4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [7.1])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} (3.4 (4 x^{3}) + \frac{d}{d x} [7.1])$

Step 3.3.5

Multiply $4$ by $3.4$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} (13.6 x^{3} + \frac{d}{d x} [7.1])$

Step 3.3.6

Since $7.1$ is constant with respect to $x$ , the derivative of $7.1$ with respect to $x$ is $0$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} (13.6 x^{3} + 0)$

Step 3.3.7

Combine fractions.

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Step 3.3.7.1

Add $13.6 x^{3}$ and $0$ .

$\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)} (13.6 x^{3})$

Step 3.3.7.2

Combine $13.6$ and $\frac{1}{(3.4 x^{4} + 7.1) (y + 3.7)}$ .

$\frac{13.6}{(3.4 x^{4} + 7.1) (y + 3.7)} x^{3}$

Step 3.3.7.3

Combine $\frac{13.6}{(3.4 x^{4} + 7.1) (y + 3.7)}$ and $x^{3}$ .

$\frac{13.6 x^{3}}{(3.4 x^{4} + 7.1) (y + 3.7)}$

Step 3.4

Simplify.

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Step 3.4.1

Factor $0.1$ out of $3.4 x^{4} + 7.1$ .

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Step 3.4.1.1

Factor $0.1$ out of $3.4 x^{4}$ .

$\frac{13.6 x^{3}}{(0.1 (34 x^{4}) + 7.1) (y + 3.7)}$

Step 3.4.1.2

Factor $0.1$ out of $7.1$ .

$\frac{13.6 x^{3}}{(0.1 (34 x^{4}) + 0.1 (71)) (y + 3.7)}$

Step 3.4.1.3

Factor $0.1$ out of $0.1 (34 x^{4}) + 0.1 (71)$ .

$\frac{13.6 x^{3}}{0.1 (34 x^{4} + 71) (y + 3.7)}$

Step 3.4.2

Factor $13.6$ out of $13.6 x^{3}$ .

$\frac{13.6 (x^{3})}{0.1 (34 x^{4} + 71) (y + 3.7)}$

Step 3.4.3

Factor $0.1$ out of $0.1 (34 x^{4} + 71) (y + 3.7)$ .

$\frac{13.6 (x^{3})}{0.1 ((34 x^{4} + 71) (y + 3.7))}$

Step 3.4.4

Separate fractions.

$\frac{13.6}{0.1} \cdot \frac{x^{3}}{(34 x^{4} + 71) (y + 3.7)}$

Step 3.4.5

Divide $13.6$ by $0.1$ .

$136 \frac{x^{3}}{(34 x^{4} + 71) (y + 3.7)}$

Step 3.4.6

Combine $136$ and $\frac{x^{3}}{(34 x^{4} + 71) (y + 3.7)}$ .

$\frac{136 x^{3}}{(34 x^{4} + 71) (y + 3.7)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$z' = \frac{136 x^{3}}{(34 x^{4} + 71) (y + 3.7)}$

Step 5

Replace $z'$ with $\frac{d z}{d x}$ .

$\frac{d z}{d x} = \frac{136 x^{3}}{(34 x^{4} + 71) (y + 3.7)}$