Calculus Examples

$f (x) = x^{\frac{4}{5}}$ , $a = 32$

Step 1

Consider the function used to find the linearization at $a$ .

$L (x) = f (a) + f' (a) (x - a)$

Step 2

Substitute the value of $a = 32$ into the linearization function.

$L (x) = f (32) + f' (32) (x - 32)$

Step 3

Evaluate $f (32)$ .

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Step 3.1

Replace the variable $x$ with $32$ in the expression.

$f (32) = {(32)}^{\frac{4}{5}}$

Step 3.2

Simplify ${(32)}^{\frac{4}{5}}$ .

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Remove parentheses.

${(32)}^{\frac{4}{5}}$

Step 3.2.1.2

Rewrite $32$ as $2^{5}$ .

${(2^{5})}^{\frac{4}{5}}$

Step 3.2.1.3

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

$2^{5 (\frac{4}{5})}$

Step 3.2.2

Cancel the common factor of $5$ .

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Step 3.2.2.1

Cancel the common factor.

$2^{5 (\frac{4}{5})}$

Step 3.2.2.2

Rewrite the expression.

$2^{4}$

Step 3.2.3

Raise $2$ to the power of $4$ .

$16$

Step 4

Find the derivative and evaluate it at $32$ .

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Step 4.1

Find the derivative of $f (x) = x^{\frac{4}{5}}$ .

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Step 4.1.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$\frac{4}{5} x^{\frac{4}{5} - 1}$

Step 4.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}}$

Step 4.1.3

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Multiply $- 1$ by $5$ .

$\frac{4}{5} x^{\frac{4 - 5}{5}}$

Step 4.1.5.2

Subtract $5$ from $4$ .

$\frac{4}{5} x^{\frac{- 1}{5}}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{4}{5} x^{- \frac{1}{5}}$

Step 4.1.7

Simplify.

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Step 4.1.7.1

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

$\frac{4}{5} \cdot \frac{1}{x^{\frac{1}{5}}}$

Step 4.1.7.2

Multiply $\frac{4}{5}$ by $\frac{1}{x^{\frac{1}{5}}}$ .

$\frac{4}{5 x^{\frac{1}{5}}}$

Step 4.2

Replace the variable $x$ with $32$ in the expression.

$\frac{4}{5 {(32)}^{\frac{1}{5}}}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the denominator.

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Step 4.3.1.1

Rewrite $32$ as $2^{5}$ .

$\frac{4}{5 \cdot {(2^{5})}^{\frac{1}{5}}}$

Step 4.3.1.2

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

$\frac{4}{5 \cdot 2^{5 (\frac{1}{5})}}$

Step 4.3.1.3

Cancel the common factor of $5$ .

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Step 4.3.1.3.1

Cancel the common factor.

$\frac{4}{5 \cdot 2^{5 (\frac{1}{5})}}$

Step 4.3.1.3.2

Rewrite the expression.

$\frac{4}{5 \cdot 2^{1}}$

Step 4.3.1.4

Evaluate the exponent.

$\frac{4}{5 \cdot 2}$

Step 4.3.2

Reduce the expression by cancelling the common factors.

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Step 4.3.2.1

Multiply $5$ by $2$ .

$\frac{4}{10}$

Step 4.3.2.2

Cancel the common factor of $4$ and $10$ .

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Step 4.3.2.2.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{10}$

Step 4.3.2.2.2

Cancel the common factors.

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Step 4.3.2.2.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 2}{2 \cdot 5}$

Step 4.3.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 5}$

Step 4.3.2.2.2.3

Rewrite the expression.

$\frac{2}{5}$

Step 5

Substitute the components into the linearization function in order to find the linearization at $a$ .

$L (x) = 16 + \frac{2}{5} (x - 32)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$L (x) = 16 + \frac{2}{5} x + \frac{2}{5} \cdot - 32$

Step 6.1.2

Combine $\frac{2}{5}$ and $x$ .

$L (x) = 16 + \frac{2 x}{5} + \frac{2}{5} \cdot - 32$

Step 6.1.3

Multiply $\frac{2}{5} \cdot - 32$ .

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Step 6.1.3.1

Combine $\frac{2}{5}$ and $- 32$ .

$L (x) = 16 + \frac{2 x}{5} + \frac{2 \cdot - 32}{5}$

Step 6.1.3.2

Multiply $2$ by $- 32$ .

$L (x) = 16 + \frac{2 x}{5} + \frac{- 64}{5}$

Step 6.1.4

Move the negative in front of the fraction.

$L (x) = 16 + \frac{2 x}{5} - \frac{64}{5}$

Step 6.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$L (x) = \frac{2 x}{5} + 16 \cdot \frac{5}{5} - \frac{64}{5}$

Step 6.3

Combine $16$ and $\frac{5}{5}$ .

$L (x) = \frac{2 x}{5} + \frac{16 \cdot 5}{5} - \frac{64}{5}$

Step 6.4

Combine the numerators over the common denominator.

$L (x) = \frac{2 x}{5} + \frac{16 \cdot 5 - 64}{5}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $16$ by $5$ .

$L (x) = \frac{2 x}{5} + \frac{80 - 64}{5}$

Step 6.5.2

Subtract $64$ from $80$ .

$L (x) = \frac{2 x}{5} + \frac{16}{5}$

Step 7