Calculus Examples

$f (x) = {(\frac{x + 3}{x - 1})}^{2}$ ; $(5, 4)$

Step 1

Find the first derivative and evaluate at $x = 5$ and $y = 4$ to find the slope of the tangent line.

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Step 1.1

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 $g (x) = \frac{x + 3}{x - 1}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $\frac{x + 3}{x - 1}$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [\frac{x + 3}{x - 1}]$

Step 1.1.2

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

$2 u \frac{d}{d x} [\frac{x + 3}{x - 1}]$

Step 1.1.3

Replace all occurrences of $u$ with $\frac{x + 3}{x - 1}$ .

$2 \frac{x + 3}{x - 1} \frac{d}{d x} [\frac{x + 3}{x - 1}]$

Step 1.2

Combine $2$ and $\frac{x + 3}{x - 1}$ .

$\frac{2 (x + 3)}{x - 1} \frac{d}{d x} [\frac{x + 3}{x - 1}]$

Step 1.3

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 $f (x) = x + 3$ and $g (x) = x - 1$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 1.4.2

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

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

Step 1.4.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{2 (x + 3)}{x - 1} \cdot \frac{(x - 1) (1 + 0) - (x + 3) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Add $1$ and $0$ .

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

Step 1.4.4.2

Multiply $x - 1$ by $1$ .

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

Step 1.4.5

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.4.6

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

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

Step 1.4.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{2 (x + 3)}{x - 1} \cdot \frac{x - 1 - (x + 3) (1 + 0)}{{(x - 1)}^{2}}$

Step 1.4.8

Combine fractions.

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Step 1.4.8.1

Add $1$ and $0$ .

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

Step 1.4.8.2

Multiply $- 1$ by $1$ .

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

Step 1.4.8.3

Multiply $\frac{2 (x + 3)}{x - 1}$ by $\frac{x - 1 - (x + 3)}{{(x - 1)}^{2}}$ .

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

Step 1.5

Multiply $x - 1$ by ${(x - 1)}^{2}$ by adding the exponents.

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Step 1.5.1

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

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

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

Step 1.5.2

Add $1$ and $2$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Simplify the numerator.

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Step 1.6.3.1

Multiply $2$ by $3$ .

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

Step 1.6.3.2

Combine the opposite terms in $x - 1 - x - 1 \cdot 3$ .

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Step 1.6.3.2.1

Subtract $x$ from $x$ .

$\frac{(2 x + 6) (0 - 1 - 1 \cdot 3)}{{(x - 1)}^{3}}$

Step 1.6.3.2.2

Subtract $1$ from $0$ .

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

Step 1.6.3.3

Multiply $- 1$ by $3$ .

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

Step 1.6.3.4

Subtract $3$ from $- 1$ .

$\frac{(2 x + 6) \cdot - 4}{{(x - 1)}^{3}}$

Step 1.6.3.5

Apply the distributive property.

$\frac{2 x \cdot - 4 + 6 \cdot - 4}{{(x - 1)}^{3}}$

Step 1.6.3.6

Multiply $- 4$ by $2$ .

$\frac{- 8 x + 6 \cdot - 4}{{(x - 1)}^{3}}$

Step 1.6.3.7

Multiply $6$ by $- 4$ .

$\frac{- 8 x - 24}{{(x - 1)}^{3}}$

Step 1.6.4

Factor $8$ out of $- 8 x - 24$ .

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Step 1.6.4.1

Factor $8$ out of $- 8 x$ .

$\frac{8 (- x) - 24}{{(x - 1)}^{3}}$

Step 1.6.4.2

Factor $8$ out of $- 24$ .

$\frac{8 (- x) + 8 (- 3)}{{(x - 1)}^{3}}$

Step 1.6.4.3

Factor $8$ out of $8 (- x) + 8 (- 3)$ .

$\frac{8 (- x - 3)}{{(x - 1)}^{3}}$

Step 1.6.5

Factor $- 1$ out of $- x$ .

$\frac{8 (- (x) - 3)}{{(x - 1)}^{3}}$

Step 1.6.6

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{8 (- (x) - 1 (3))}{{(x - 1)}^{3}}$

Step 1.6.7

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{8 (- (x + 3))}{{(x - 1)}^{3}}$

Step 1.6.8

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{8 (- 1 (x + 3))}{{(x - 1)}^{3}}$

Step 1.6.9

Move the negative in front of the fraction.

$- \frac{8 (x + 3)}{{(x - 1)}^{3}}$

Step 1.7

Evaluate the derivative at $x = 5$ .

$- \frac{8 ((5) + 3)}{{((5) - 1)}^{3}}$

Step 1.8

Simplify.

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Step 1.8.1

Add $5$ and $3$ .

$- \frac{8 \cdot 8}{{(5 - 1)}^{3}}$

Step 1.8.2

Simplify the denominator.

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Step 1.8.2.1

Subtract $1$ from $5$ .

$- \frac{8 \cdot 8}{4^{3}}$

Step 1.8.2.2

Raise $4$ to the power of $3$ .

$- \frac{8 \cdot 8}{64}$

Step 1.8.3

Reduce the expression by cancelling the common factors.

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Step 1.8.3.1

Multiply $8$ by $8$ .

$- \frac{64}{64}$

Step 1.8.3.2

Cancel the common factor of $64$ .

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Step 1.8.3.2.1

Cancel the common factor.

$- \frac{64}{64}$

Step 1.8.3.2.2

Rewrite the expression.

$- 1 \cdot 1$

Step 1.8.3.3

Multiply $- 1$ by $1$ .

$- 1$

Step 2

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

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Step 2.1

Use the slope $- 1$ and a given point $(5, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - 1 \cdot (x - (5))$

Step 2.2

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

$y - 4 = - 1 \cdot (x - 5)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 1 \cdot (x - 5)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 - 1 \cdot (x - 5)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = - 1 \cdot (x - 5)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = - 1 x - 1 \cdot - 5$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

Rewrite $- 1 x$ as $- x$ .

$y - 4 = - x - 1 \cdot - 5$

Step 2.3.1.4.2

Multiply $- 1$ by $- 5$ .

$y - 4 = - x + 5$

Step 2.3.2

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

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Step 2.3.2.1

Add $4$ to both sides of the equation.

$y = - x + 5 + 4$

Step 2.3.2.2

Add $5$ and $4$ .

$y = - x + 9$

Step 3