Calculus Examples

$f (x) = \sqrt{25 - x^{2}}$ at the point where $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

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Step 1.1

Substitute $4$ in for $x$ .

$y = \sqrt{25 - {(4)}^{2}}$

Step 1.2

Simplify $\sqrt{25 - {(4)}^{2}}$ .

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Step 1.2.1

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

$y = \sqrt{25 - 1 \cdot 16}$

Step 1.2.2

Multiply $- 1$ by $16$ .

$y = \sqrt{25 - 16}$

Step 1.2.3

Subtract $16$ from $25$ .

$y = \sqrt{9}$

Step 1.2.4

Rewrite $9$ as $3^{2}$ .

$y = \sqrt{3^{2}}$

Step 1.2.5

Pull terms out from under the radical, assuming positive real numbers.

$y = 3$

Step 2

Find the first derivative and evaluate at $x = 4$ and $y = 3$ to find the slope of the tangent line.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 - x^{2}}$ as ${(25 - x^{2})}^{\frac{1}{2}}$ .

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

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = 25 - x^{2}$ .

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Step 2.2.1

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

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [25 - x^{2}]$

Step 2.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [25 - x^{2}]$

Step 2.2.3

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

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

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

Combine $\frac{1}{2}$ and ${(25 - x^{2})}^{- \frac{1}{2}}$ .

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

Step 2.7.3

Move ${(25 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8

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

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

Step 2.9

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

$\frac{1}{2 {(25 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

$\frac{1}{2 {(25 - x^{2})}^{\frac{1}{2}}} (- (2 x))$

Step 2.13

Simplify terms.

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Step 2.13.1

Multiply $2$ by $- 1$ .

$\frac{1}{2 {(25 - x^{2})}^{\frac{1}{2}}} (- 2 x)$

Step 2.13.2

Combine $- 2$ and $\frac{1}{2 {(25 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{- 2}{2 {(25 - x^{2})}^{\frac{1}{2}}} x$

Step 2.13.3

Combine $\frac{- 2}{2 {(25 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{- 2 x}{2 {(25 - x^{2})}^{\frac{1}{2}}}$

Step 2.13.4

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x)}{2 {(25 - x^{2})}^{\frac{1}{2}}}$

Step 2.14

Cancel the common factors.

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Step 2.14.1

Factor $2$ out of $2 {(25 - x^{2})}^{\frac{1}{2}}$ .

$\frac{2 (- x)}{2 ({(25 - x^{2})}^{\frac{1}{2}})}$

Step 2.14.2

Cancel the common factor.

$\frac{2 (- x)}{2 {(25 - x^{2})}^{\frac{1}{2}}}$

Step 2.14.3

Rewrite the expression.

$\frac{- x}{{(25 - x^{2})}^{\frac{1}{2}}}$

Step 2.15

Move the negative in front of the fraction.

$- \frac{x}{{(25 - x^{2})}^{\frac{1}{2}}}$

Step 2.16

Evaluate the derivative at $x = 4$ .

$- \frac{4}{{(25 - {(4)}^{2})}^{\frac{1}{2}}}$

Step 2.17

Simplify the denominator.

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Step 2.17.1

Simplify each term.

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Step 2.17.1.1

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

$- \frac{4}{{(25 - 1 \cdot 16)}^{\frac{1}{2}}}$

Step 2.17.1.2

Multiply $- 1$ by $16$ .

$- \frac{4}{{(25 - 16)}^{\frac{1}{2}}}$

Step 2.17.2

Subtract $16$ from $25$ .

$- \frac{4}{9^{\frac{1}{2}}}$

Step 2.17.3

Rewrite $9$ as $3^{2}$ .

$- \frac{4}{{(3^{2})}^{\frac{1}{2}}}$

Step 2.17.4

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

$- \frac{4}{3^{2 (\frac{1}{2})}}$

Step 2.17.5

Cancel the common factor of $2$ .

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Step 2.17.5.1

Cancel the common factor.

$- \frac{4}{3^{2 (\frac{1}{2})}}$

Step 2.17.5.2

Rewrite the expression.

$- \frac{4}{3^{1}}$

Step 2.17.6

Evaluate the exponent.

$- \frac{4}{3}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- \frac{4}{3}$ and a given point $(4, 3)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{4}{3} \cdot (x - (4))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 3 = - \frac{4}{3} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{4}{3} \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{4}{3} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 3 = - \frac{4}{3} \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - 3 = - \frac{4}{3} x - \frac{4}{3} \cdot - 4$

Step 3.3.1.4

Combine $x$ and $\frac{4}{3}$ .

$y - 3 = - \frac{x \cdot 4}{3} - \frac{4}{3} \cdot - 4$

Step 3.3.1.5

Multiply $- \frac{4}{3} \cdot - 4$ .

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Step 3.3.1.5.1

Multiply $- 4$ by $- 1$ .

$y - 3 = - \frac{x \cdot 4}{3} + 4 (\frac{4}{3})$

Step 3.3.1.5.2

Combine $4$ and $\frac{4}{3}$ .

$y - 3 = - \frac{x \cdot 4}{3} + \frac{4 \cdot 4}{3}$

Step 3.3.1.5.3

Multiply $4$ by $4$ .

$y - 3 = - \frac{x \cdot 4}{3} + \frac{16}{3}$

Step 3.3.1.6

Move $4$ to the left of $x$ .

$y - 3 = - \frac{4 x}{3} + \frac{16}{3}$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $3$ to both sides of the equation.

$y = - \frac{4 x}{3} + \frac{16}{3} + 3$

Step 3.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{4 x}{3} + \frac{16}{3} + 3 \cdot \frac{3}{3}$

Step 3.3.2.3

Combine $3$ and $\frac{3}{3}$ .

$y = - \frac{4 x}{3} + \frac{16}{3} + \frac{3 \cdot 3}{3}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{4 x}{3} + \frac{16 + 3 \cdot 3}{3}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $3$ by $3$ .

$y = - \frac{4 x}{3} + \frac{16 + 9}{3}$

Step 3.3.2.5.2

Add $16$ and $9$ .

$y = - \frac{4 x}{3} + \frac{25}{3}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{4}{3} x) + \frac{25}{3}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{4}{3} x + \frac{25}{3}$

Step 4