Calculus Examples

$\frac{f (x) \cdot 1}{\sqrt{3 x - 2}}$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Multiply $f (x)$ by $1$ .

$\frac{d}{d f} [\frac{f (x)}{\sqrt{3 x - 2}}]$

Step 3

Multiply $\frac{f (x)}{\sqrt{3 x - 2}}$ by $\frac{\sqrt{3 x - 2}}{\sqrt{3 x - 2}}$ .

$\frac{d}{d f} [\frac{f (x)}{\sqrt{3 x - 2}} \cdot \frac{\sqrt{3 x - 2}}{\sqrt{3 x - 2}}]$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{f (x)}{\sqrt{3 x - 2}}$ by $\frac{\sqrt{3 x - 2}}{\sqrt{3 x - 2}}$ .

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{\sqrt{3 x - 2} \sqrt{3 x - 2}}]$

Step 4.2

Raise $\sqrt{3 x - 2}$ to the power of $1$ .

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{\sqrt{3 x - 2}}^{1} \sqrt{3 x - 2}}]$

Step 4.3

Raise $\sqrt{3 x - 2}$ to the power of $1$ .

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{\sqrt{3 x - 2}}^{1} {\sqrt{3 x - 2}}^{1}}]$

Step 4.4

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

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{\sqrt{3 x - 2}}^{1 + 1}}]$

Step 4.5

Add $1$ and $1$ .

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{\sqrt{3 x - 2}}^{2}}]$

Step 4.6

Rewrite ${\sqrt{3 x - 2}}^{2}$ as $3 x - 2$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 2}$ as ${(3 x - 2)}^{\frac{1}{2}}$ .

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{({(3 x - 2)}^{\frac{1}{2}})}^{2}}]$

Step 4.6.2

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

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{(3 x - 2)}^{\frac{1}{2} \cdot 2}}]$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{(3 x - 2)}^{\frac{2}{2}}}]$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{(3 x - 2)}^{\frac{2}{2}}}]$

Step 4.6.4.2

Rewrite the expression.

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{{(3 x - 2)}^{1}}]$

Step 4.6.5

Simplify.

$\frac{d}{d f} [\frac{f (x) \sqrt{3 x - 2}}{3 x - 2}]$

Step 5

Since $\frac{f (x) \sqrt{3 x - 2}}{3 x - 2}$ is constant with respect to $f$ , the derivative of $\frac{f (x) \sqrt{3 x - 2}}{3 x - 2}$ with respect to $f$ is $0$ .

$0$