Calculus Examples

$f (x) = \frac{x^{2} + 5 x}{25 - x^{2}}$

Step 1

Find the first derivative.

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Step 1.1

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 $f (x) = x^{2} + 5 x$ and $g (x) = 25 - x^{2}$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

$\frac{(25 - x^{2}) (2 x + 5 \cdot 1) - (x^{2} + 5 x) \frac{d}{d x} [25 - x^{2}]}{{(25 - x^{2})}^{2}}$

Step 1.2.5

Multiply $5$ by $1$ .

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

Step 1.2.6

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

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

Step 1.2.7

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

$\frac{(25 - x^{2}) (2 x + 5) - (x^{2} + 5 x) (0 + \frac{d}{d x} [- x^{2}])}{{(25 - x^{2})}^{2}}$

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Multiply.

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Step 1.2.10.1

Multiply $- 1$ by $- 1$ .

$\frac{(25 - x^{2}) (2 x + 5) + 1 (x^{2} + 5 x) \frac{d}{d x} [x^{2}]}{{(25 - x^{2})}^{2}}$

Step 1.2.10.2

Multiply $x^{2} + 5 x$ by $1$ .

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

Step 1.2.11

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

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

Step 1.2.12

Move $2$ to the left of $x^{2} + 5 x$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Expand $(25 - x^{2}) (2 x + 5)$ using the FOIL Method.

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Step 1.3.3.1.1.1

Apply the distributive property.

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

Step 1.3.3.1.1.2

Apply the distributive property.

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

Step 1.3.3.1.1.3

Apply the distributive property.

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

Step 1.3.3.1.2

Simplify each term.

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Step 1.3.3.1.2.1

Multiply $2$ by $25$ .

$\frac{50 x + 25 \cdot 5 - x^{2} (2 x) - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.2

Multiply $25$ by $5$ .

$\frac{50 x + 125 - x^{2} (2 x) - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.3

Rewrite using the commutative property of multiplication.

$\frac{50 x + 125 - 1 \cdot 2 x^{2} x - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.3.1.2.4.1

Move $x$ .

$\frac{50 x + 125 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.4.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.3.1.2.4.2.1

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

$\frac{50 x + 125 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.4.2.2

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

$\frac{50 x + 125 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.4.3

Add $1$ and $2$ .

$\frac{50 x + 125 - 1 \cdot 2 x^{3} - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.5

Multiply $- 1$ by $2$ .

$\frac{50 x + 125 - 2 x^{3} - x^{2} \cdot 5 + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.2.6

Multiply $5$ by $- 1$ .

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{2} x + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.3.1.3.1

Move $x$ .

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 (x \cdot x^{2}) + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.3.1.3.2.1

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

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 (x^{1} x^{2}) + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.3.2.2

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

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{1 + 2} + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.3.3

Add $1$ and $2$ .

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{3} + 2 (5 x) x}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.3.1.4.1

Move $x$ .

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{3} + 2 (5 (x \cdot x))}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.4.2

Multiply $x$ by $x$ .

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{3} + 2 (5 x^{2})}{{(25 - x^{2})}^{2}}$

Step 1.3.3.1.5

Multiply $5$ by $2$ .

$\frac{50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{3} + 10 x^{2}}{{(25 - x^{2})}^{2}}$

Step 1.3.3.2

Combine the opposite terms in $50 x + 125 - 2 x^{3} - 5 x^{2} + 2 x^{3} + 10 x^{2}$ .

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Step 1.3.3.2.1

Add $- 2 x^{3}$ and $2 x^{3}$ .

$\frac{50 x + 125 - 5 x^{2} + 0 + 10 x^{2}}{{(25 - x^{2})}^{2}}$

Step 1.3.3.2.2

Add $50 x + 125 - 5 x^{2}$ and $0$ .

$\frac{50 x + 125 - 5 x^{2} + 10 x^{2}}{{(25 - x^{2})}^{2}}$

Step 1.3.3.3

Add $- 5 x^{2}$ and $10 x^{2}$ .

$\frac{50 x + 125 + 5 x^{2}}{{(25 - x^{2})}^{2}}$

Step 1.3.4

Reorder terms.

$\frac{5 x^{2} + 50 x + 125}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Factor $5$ out of $5 x^{2} + 50 x + 125$ .

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Step 1.3.5.1.1

Factor $5$ out of $5 x^{2}$ .

$\frac{5 (x^{2}) + 50 x + 125}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5.1.2

Factor $5$ out of $50 x$ .

$\frac{5 (x^{2}) + 5 (10 x) + 125}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5.1.3

Factor $5$ out of $125$ .

$\frac{5 x^{2} + 5 (10 x) + 5 \cdot 25}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5.1.4

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

$\frac{5 (x^{2} + 10 x) + 5 \cdot 25}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5.1.5

Factor $5$ out of $5 (x^{2} + 10 x) + 5 \cdot 25$ .

$\frac{5 (x^{2} + 10 x + 25)}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5.2

Factor using the perfect square rule.

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Step 1.3.5.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{5 (x^{2} + 10 x + 5^{2})}{{(- x^{2} + 25)}^{2}}$

Step 1.3.5.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 1.3.5.2.3

Rewrite the polynomial.

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

Step 1.3.5.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

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

Step 1.3.6

Simplify the denominator.

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Step 1.3.6.1

Rewrite $25$ as $5^{2}$ .

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

Step 1.3.6.2

Reorder $- x^{2}$ and $5^{2}$ .

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

Step 1.3.6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

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

Step 1.3.6.4

Apply the product rule to $(5 + x) (5 - x)$ .

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

Step 1.3.7

Cancel the common factor of ${(x + 5)}^{2}$ and ${(5 + x)}^{2}$ .

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Step 1.3.7.1

Reorder terms.

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

Step 1.3.7.2

Cancel the common factor.

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

Step 1.3.7.3

Rewrite the expression.

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{{(5 - x)}^{2}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{{(5 - x)}^{2}}]$ .

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(5 - x)}^{2}}$ as ${({(5 - x)}^{2})}^{- 1}$ .

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

Step 2.1.2.2

Multiply the exponents in ${({(5 - x)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $5$ .

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

Step 2.3.2

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

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

Step 2.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$- 10 {(5 - x)}^{- 3} (0 + \frac{d}{d x} [- x])$

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Multiply $- 1$ by $- 10$ .

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

Step 2.3.7

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

$10 {(5 - x)}^{- 3} \cdot 1$

Step 2.3.8

Multiply $10$ by $1$ .

$10 {(5 - x)}^{- 3}$

Step 2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 \frac{1}{{(5 - x)}^{3}}$

Step 2.5

Simplify.

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Step 2.5.1

Combine $10$ and $\frac{1}{{(5 - x)}^{3}}$ .

$\frac{10}{{(5 - x)}^{3}}$

Step 2.5.2

Reorder terms.

$f'' (x) = \frac{10}{{(- x + 5)}^{3}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{10}{{(- x + 5)}^{3}}$ .

$\frac{10}{{(- x + 5)}^{3}}$