Trigonometry Examples

${(5 x^{2} - 7 \ln (x))}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 5 x^{2} - 7 \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $5 x^{2} - 7 \ln (x)$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [5 x^{2} - 7 \ln (x)]$

Step 1.2

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

$2 u \frac{d}{d x} [5 x^{2} - 7 \ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $5 x^{2} - 7 \ln (x)$ .

$2 (5 x^{2} - 7 \ln (x)) \frac{d}{d x} [5 x^{2} - 7 \ln (x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{2} - 7 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 7 \ln (x)]$ .

$2 (5 x^{2} - 7 \ln (x)) (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 7 \ln (x)])$

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$2 (5 x^{2} - 7 \ln (x)) (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 \ln (x)])$

Step 2.3

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

$2 (5 x^{2} - 7 \ln (x)) (5 (2 x) + \frac{d}{d x} [- 7 \ln (x)])$

Step 2.4

Multiply $2$ by $5$ .

$2 (5 x^{2} - 7 \ln (x)) (10 x + \frac{d}{d x} [- 7 \ln (x)])$

Step 2.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 \ln (x)$ with respect to $x$ is $- 7 \frac{d}{d x} [\ln (x)]$ .

$2 (5 x^{2} - 7 \ln (x)) (10 x - 7 \frac{d}{d x} [\ln (x)])$

Step 3

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

$2 (5 x^{2} - 7 \ln (x)) (10 x - 7 \frac{1}{x})$

Step 4

Combine fractions.

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Step 4.1

Combine $- 7$ and $\frac{1}{x}$ .

$2 (5 x^{2} - 7 \ln (x)) (10 x + \frac{- 7}{x})$

Step 4.2

Move the negative in front of the fraction.

$2 (5 x^{2} - 7 \ln (x)) (10 x - \frac{7}{x})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$(2 (5 x^{2}) + 2 (- 7 \ln (x))) (10 x - \frac{7}{x})$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $5$ by $2$ .

$(10 x^{2} + 2 (- 7 \ln (x))) (10 x - \frac{7}{x})$

Step 5.2.2

Multiply $- 7$ by $2$ .

$(10 x^{2} - 14 \ln (x)) (10 x - \frac{7}{x})$

Step 5.3

Reorder the factors of $(10 x^{2} - 14 \ln (x)) (10 x - \frac{7}{x})$ .

$(10 x - \frac{7}{x}) (10 x^{2} - 14 \ln (x))$