Calculus Examples

$\frac{54 x}{{(25 - x^{2})}^{2}}$

Step 1

Since $54$ is constant with respect to $x$ , the derivative of $\frac{54 x}{{(25 - x^{2})}^{2}}$ with respect to $x$ is $54 \frac{d}{d x} [\frac{x}{{(25 - x^{2})}^{2}}]$ .

$54 \frac{d}{d x} [\frac{x}{{(25 - x^{2})}^{2}}]$

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 $g (x) = {(25 - x^{2})}^{2}$ .

$54 \frac{{(25 - x^{2})}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(25 - x^{2})}^{2}]}{{({(25 - x^{2})}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(25 - x^{2})}^{2})}^{2}$ .

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Step 3.1.1

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

$54 \frac{{(25 - x^{2})}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(25 - x^{2})}^{2}]}{{(25 - x^{2})}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$54 \frac{{(25 - x^{2})}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(25 - x^{2})}^{2}]}{{(25 - x^{2})}^{4}}$

Step 3.2

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

$54 \frac{{(25 - x^{2})}^{2} \cdot 1 - x \frac{d}{d x} [{(25 - x^{2})}^{2}]}{{(25 - x^{2})}^{4}}$

Step 3.3

Multiply ${(25 - x^{2})}^{2}$ by $1$ .

$54 \frac{{(25 - x^{2})}^{2} - x \frac{d}{d x} [{(25 - x^{2})}^{2}]}{{(25 - x^{2})}^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 25 - x^{2}$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $25 - x^{2}$ .

$54 \frac{{(25 - x^{2})}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [25 - x^{2}])}{{(25 - x^{2})}^{4}}$

Step 4.2

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

$54 \frac{{(25 - x^{2})}^{2} - x (2 u \frac{d}{d x} [25 - x^{2}])}{{(25 - x^{2})}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $25 - x^{2}$ .

$54 \frac{{(25 - x^{2})}^{2} - x (2 (25 - x^{2}) \frac{d}{d x} [25 - x^{2}])}{{(25 - x^{2})}^{4}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

$54 \frac{{(25 - x^{2})}^{2} - 2 x ((25 - x^{2}) \frac{d}{d x} [25 - x^{2}])}{{(25 - x^{2})}^{4}}$

Step 5.2

Factor $25 - x^{2}$ out of ${(25 - x^{2})}^{2} - 2 x ((25 - x^{2}) \frac{d}{d x} [25 - x^{2}])$ .

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Step 5.2.1

Factor $25 - x^{2}$ out of ${(25 - x^{2})}^{2}$ .

$54 \frac{(25 - x^{2}) (25 - x^{2}) - 2 x ((25 - x^{2}) \frac{d}{d x} [25 - x^{2}])}{{(25 - x^{2})}^{4}}$

Step 5.2.2

Factor $25 - x^{2}$ out of $- 2 x ((25 - x^{2}) \frac{d}{d x} [25 - x^{2}])$ .

$54 \frac{(25 - x^{2}) (25 - x^{2}) + (25 - x^{2}) (- 2 x (\frac{d}{d x} [25 - x^{2}]))}{{(25 - x^{2})}^{4}}$

Step 5.2.3

Factor $25 - x^{2}$ out of $(25 - x^{2}) (25 - x^{2}) + (25 - x^{2}) (- 2 x (\frac{d}{d x} [25 - x^{2}]))$ .

$54 \frac{(25 - x^{2}) (25 - x^{2} - 2 x (\frac{d}{d x} [25 - x^{2}]))}{{(25 - x^{2})}^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $25 - x^{2}$ out of ${(25 - x^{2})}^{4}$ .

$54 \frac{(25 - x^{2}) (25 - x^{2} - 2 x (\frac{d}{d x} [25 - x^{2}]))}{(25 - x^{2}) {(25 - x^{2})}^{3}}$

Step 6.2

Cancel the common factor.

$54 \frac{(25 - x^{2}) (25 - x^{2} - 2 x (\frac{d}{d x} [25 - x^{2}]))}{(25 - x^{2}) {(25 - x^{2})}^{3}}$

Step 6.3

Rewrite the expression.

$54 \frac{25 - x^{2} - 2 x (\frac{d}{d x} [25 - x^{2}])}{{(25 - x^{2})}^{3}}$

Step 7

By the Sum Rule, the derivative of $25 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$ .

$54 \frac{25 - x^{2} - 2 x (\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}])}{{(25 - x^{2})}^{3}}$

Step 8

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$54 \frac{25 - x^{2} - 2 x (0 + \frac{d}{d x} [- x^{2}])}{{(25 - x^{2})}^{3}}$

Step 9

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$54 \frac{25 - x^{2} - 2 x \frac{d}{d x} [- x^{2}]}{{(25 - x^{2})}^{3}}$

Step 10

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$54 \frac{25 - x^{2} - 2 x (- \frac{d}{d x} [x^{2}])}{{(25 - x^{2})}^{3}}$

Step 11

Multiply $- 1$ by $- 2$ .

$54 \frac{25 - x^{2} + 2 x \frac{d}{d x} [x^{2}]}{{(25 - x^{2})}^{3}}$

Step 12

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

$54 \frac{25 - x^{2} + 2 x (2 x)}{{(25 - x^{2})}^{3}}$

Step 13

Multiply $2$ by $2$ .

$54 \frac{25 - x^{2} + 4 x \cdot x}{{(25 - x^{2})}^{3}}$

Step 14

Raise $x$ to the power of $1$ .

$54 \frac{25 - x^{2} + 4 (x^{1} x)}{{(25 - x^{2})}^{3}}$

Step 15

Raise $x$ to the power of $1$ .

$54 \frac{25 - x^{2} + 4 (x^{1} x^{1})}{{(25 - x^{2})}^{3}}$

Step 16

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

$54 \frac{25 - x^{2} + 4 x^{1 + 1}}{{(25 - x^{2})}^{3}}$

Step 17

Add $1$ and $1$ .

$54 \frac{25 - x^{2} + 4 x^{2}}{{(25 - x^{2})}^{3}}$

Step 18

Add $- x^{2}$ and $4 x^{2}$ .

$54 \frac{25 + 3 x^{2}}{{(25 - x^{2})}^{3}}$

Step 19

Combine $54$ and $\frac{25 + 3 x^{2}}{{(25 - x^{2})}^{3}}$ .

$\frac{54 (25 + 3 x^{2})}{{(25 - x^{2})}^{3}}$

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

$\frac{54 \cdot 25 + 54 (3 x^{2})}{{(25 - x^{2})}^{3}}$

Step 20.2

Simplify each term.

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Step 20.2.1

Multiply $54$ by $25$ .

$\frac{1350 + 54 (3 x^{2})}{{(25 - x^{2})}^{3}}$

Step 20.2.2

Multiply $3$ by $54$ .

$\frac{1350 + 162 x^{2}}{{(25 - x^{2})}^{3}}$

Step 20.3

Reorder terms.

$\frac{162 x^{2} + 1350}{{(- x^{2} + 25)}^{3}}$