Calculus Examples

$f (x) = \frac{3 x}{{(2 - 5 x)}^{5}}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 x}{{(2 - 5 x)}^{5}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{x}{{(2 - 5 x)}^{5}}]$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(2 - 5 x)}^{5}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(2 - 5 x)}^{5})}^{2}$ .

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Step 3.1.1

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

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

Step 3.1.2

Multiply $5$ by $2$ .

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

Step 3.2

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

$3 \frac{{(2 - 5 x)}^{5} \cdot 1 - x \frac{d}{d x} [{(2 - 5 x)}^{5}]}{{(2 - 5 x)}^{10}}$

Step 3.3

Multiply ${(2 - 5 x)}^{5}$ by $1$ .

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

Step 4

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

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Step 4.1

To apply the Chain Rule, set $u$ as $2 - 5 x$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$3 \frac{{(2 - 5 x)}^{5} - x (5 u^{4} \frac{d}{d x} [2 - 5 x])}{{(2 - 5 x)}^{10}}$

Step 4.3

Replace all occurrences of $u$ with $2 - 5 x$ .

$3 \frac{{(2 - 5 x)}^{5} - x (5 {(2 - 5 x)}^{4} \frac{d}{d x} [2 - 5 x])}{{(2 - 5 x)}^{10}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $5$ by $- 1$ .

$3 \frac{{(2 - 5 x)}^{5} - 5 x ({(2 - 5 x)}^{4} \frac{d}{d x} [2 - 5 x])}{{(2 - 5 x)}^{10}}$

Step 5.2

Factor ${(2 - 5 x)}^{4}$ out of ${(2 - 5 x)}^{5} - 5 x ({(2 - 5 x)}^{4} \frac{d}{d x} [2 - 5 x])$ .

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Step 5.2.1

Factor ${(2 - 5 x)}^{4}$ out of ${(2 - 5 x)}^{5}$ .

$3 \frac{{(2 - 5 x)}^{4} (2 - 5 x) - 5 x ({(2 - 5 x)}^{4} \frac{d}{d x} [2 - 5 x])}{{(2 - 5 x)}^{10}}$

Step 5.2.2

Factor ${(2 - 5 x)}^{4}$ out of $- 5 x ({(2 - 5 x)}^{4} \frac{d}{d x} [2 - 5 x])$ .

$3 \frac{{(2 - 5 x)}^{4} (2 - 5 x) + {(2 - 5 x)}^{4} (- 5 x (\frac{d}{d x} [2 - 5 x]))}{{(2 - 5 x)}^{10}}$

Step 5.2.3

Factor ${(2 - 5 x)}^{4}$ out of ${(2 - 5 x)}^{4} (2 - 5 x) + {(2 - 5 x)}^{4} (- 5 x (\frac{d}{d x} [2 - 5 x]))$ .

$3 \frac{{(2 - 5 x)}^{4} (2 - 5 x - 5 x (\frac{d}{d x} [2 - 5 x]))}{{(2 - 5 x)}^{10}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor ${(2 - 5 x)}^{4}$ out of ${(2 - 5 x)}^{10}$ .

$3 \frac{{(2 - 5 x)}^{4} (2 - 5 x - 5 x (\frac{d}{d x} [2 - 5 x]))}{{(2 - 5 x)}^{4} {(2 - 5 x)}^{6}}$

Step 6.2

Cancel the common factor.

$3 \frac{{(2 - 5 x)}^{4} (2 - 5 x - 5 x (\frac{d}{d x} [2 - 5 x]))}{{(2 - 5 x)}^{4} {(2 - 5 x)}^{6}}$

Step 6.3

Rewrite the expression.

$3 \frac{2 - 5 x - 5 x (\frac{d}{d x} [2 - 5 x])}{{(2 - 5 x)}^{6}}$

Step 7

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

$3 \frac{2 - 5 x - 5 x (\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x])}{{(2 - 5 x)}^{6}}$

Step 8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 \frac{2 - 5 x - 5 x (0 + \frac{d}{d x} [- 5 x])}{{(2 - 5 x)}^{6}}$

Step 9

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

$3 \frac{2 - 5 x - 5 x \frac{d}{d x} [- 5 x]}{{(2 - 5 x)}^{6}}$

Step 10

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

$3 \frac{2 - 5 x - 5 x (- 5 \frac{d}{d x} [x])}{{(2 - 5 x)}^{6}}$

Step 11

Multiply $- 5$ by $- 5$ .

$3 \frac{2 - 5 x + 25 x \frac{d}{d x} [x]}{{(2 - 5 x)}^{6}}$

Step 12

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

$3 \frac{2 - 5 x + 25 x \cdot 1}{{(2 - 5 x)}^{6}}$

Step 13

Simplify terms.

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Step 13.1

Multiply $25$ by $1$ .

$3 \frac{2 - 5 x + 25 x}{{(2 - 5 x)}^{6}}$

Step 13.2

Add $- 5 x$ and $25 x$ .

$3 \frac{2 + 20 x}{{(2 - 5 x)}^{6}}$

Step 13.3

Combine $3$ and $\frac{2 + 20 x}{{(2 - 5 x)}^{6}}$ .

$\frac{3 (2 + 20 x)}{{(2 - 5 x)}^{6}}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{3 \cdot 2 + 3 (20 x)}{{(2 - 5 x)}^{6}}$

Step 14.2

Simplify each term.

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Step 14.2.1

Multiply $3$ by $2$ .

$\frac{6 + 3 (20 x)}{{(2 - 5 x)}^{6}}$

Step 14.2.2

Multiply $20$ by $3$ .

$\frac{6 + 60 x}{{(2 - 5 x)}^{6}}$

Step 14.3

Reorder terms.

$\frac{60 x + 6}{{(- 5 x + 2)}^{6}}$