Calculus Examples

$f (x) = \frac{3}{{(x^{2} - 3 x)}^{2}}$ , $(4, \frac{3}{16})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = \frac{3}{16}$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{{(x^{2} - 3 x)}^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{{(x^{2} - 3 x)}^{2}}]$ .

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

Step 1.1.2

Apply basic rules of exponents.

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Step 1.1.2.1

Rewrite $\frac{1}{{(x^{2} - 3 x)}^{2}}$ as ${({(x^{2} - 3 x)}^{2})}^{- 1}$ .

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

Step 1.1.2.2

Multiply the exponents in ${({(x^{2} - 3 x)}^{2})}^{- 1}$ .

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2

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 $g (x) = x^{2} - 3 x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 3 x$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} - 3 x$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 2$ by $3$ .

$- 6 ({(x^{2} - 3 x)}^{- 3} \frac{d}{d x} [x^{2} - 3 x])$

Step 1.3.2

By the Sum Rule, the derivative of $x^{2} - 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]$ .

$- 6 {(x^{2} - 3 x)}^{- 3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x])$

Step 1.3.3

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

$- 6 {(x^{2} - 3 x)}^{- 3} (2 x + \frac{d}{d x} [- 3 x])$

Step 1.3.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 6 {(x^{2} - 3 x)}^{- 3} (2 x - 3 \frac{d}{d x} [x])$

Step 1.3.5

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

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

Step 1.3.6

Multiply $- 3$ by $1$ .

$- 6 {(x^{2} - 3 x)}^{- 3} (2 x - 3)$

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Combine $- 6$ and $\frac{1}{{(x^{2} - 3 x)}^{3}}$ .

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

Step 1.4.2.2

Move the negative in front of the fraction.

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

Step 1.4.3

Reorder the factors of $- \frac{6}{{(x^{2} - 3 x)}^{3}} (2 x - 3)$ .

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

Step 1.4.4

Apply the distributive property.

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

Step 1.4.5

Multiply $2$ by $- 1$ .

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

Step 1.4.6

Multiply $- 1$ by $- 3$ .

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

Step 1.4.7

Simplify the denominator.

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Step 1.4.7.1

Factor $x$ out of $x^{2} - 3 x$ .

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Step 1.4.7.1.1

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

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

Step 1.4.7.1.2

Factor $x$ out of $- 3 x$ .

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

Step 1.4.7.1.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

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

Step 1.4.7.2

Apply the product rule to $x (x - 3)$ .

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

Step 1.4.8

Multiply $- 2 x + 3$ by $\frac{6}{x^{3} {(x - 3)}^{3}}$ .

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

Step 1.4.9

Move $6$ to the left of $- 2 x + 3$ .

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

Step 1.4.10

Factor $- 1$ out of $- 2 x$ .

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

Step 1.4.11

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

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

Step 1.4.12

Factor $- 1$ out of $- (2 x) - 1 (- 3)$ .

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

Step 1.4.13

Rewrite $- (2 x - 3)$ as $- 1 (2 x - 3)$ .

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

Step 1.4.14

Move the negative in front of the fraction.

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

Step 1.5

Evaluate the derivative at $x = 4$ .

$- \frac{6 (2 (4) - 3)}{{(4)}^{3} {((4) - 3)}^{3}}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Multiply $2$ by $4$ .

$- \frac{6 (8 - 3)}{4^{3} {(4 - 3)}^{3}}$

Step 1.6.1.2

Subtract $3$ from $8$ .

$- \frac{6 \cdot 5}{4^{3} {(4 - 3)}^{3}}$

Step 1.6.2

Simplify the denominator.

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Step 1.6.2.1

Subtract $3$ from $4$ .

$- \frac{6 \cdot 5}{4^{3} \cdot 1^{3}}$

Step 1.6.2.2

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

$- \frac{6 \cdot 5}{64 \cdot 1^{3}}$

Step 1.6.2.3

One to any power is one.

$- \frac{6 \cdot 5}{64 \cdot 1}$

Step 1.6.3

Reduce the expression by cancelling the common factors.

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Step 1.6.3.1

Multiply $6$ by $5$ .

$- \frac{30}{64 \cdot 1}$

Step 1.6.3.2

Multiply $64$ by $1$ .

$- \frac{30}{64}$

Step 1.6.3.3

Cancel the common factor of $30$ and $64$ .

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Step 1.6.3.3.1

Factor $2$ out of $30$ .

$- \frac{2 (15)}{64}$

Step 1.6.3.3.2

Cancel the common factors.

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Step 1.6.3.3.2.1

Factor $2$ out of $64$ .

$- \frac{2 \cdot 15}{2 \cdot 32}$

Step 1.6.3.3.2.2

Cancel the common factor.

$- \frac{2 \cdot 15}{2 \cdot 32}$

Step 1.6.3.3.2.3

Rewrite the expression.

$- \frac{15}{32}$

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 $- \frac{15}{32}$ and a given point $(4, \frac{3}{16})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{3}{16}) = - \frac{15}{32} \cdot (x - (4))$

Step 2.2

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

$y - \frac{3}{16} = - \frac{15}{32} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{15}{32} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{3}{16} = 0 + 0 - \frac{15}{32} \cdot (x - 4)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - \frac{3}{16} = - \frac{15}{32} x - \frac{15}{32} \cdot - 4$

Step 2.3.1.2.2

Combine $x$ and $\frac{15}{32}$ .

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} - \frac{15}{32} \cdot - 4$

Step 2.3.1.2.3

Cancel the common factor of $4$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{15}{32}$ into the numerator.

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{- 15}{32} \cdot - 4$

Step 2.3.1.2.3.2

Factor $4$ out of $32$ .

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{- 15}{4 (8)} \cdot - 4$

Step 2.3.1.2.3.3

Factor $4$ out of $- 4$ .

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{- 15}{4 \cdot 8} \cdot (4 \cdot - 1)$

Step 2.3.1.2.3.4

Cancel the common factor.

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{- 15}{4 \cdot 8} \cdot (4 \cdot - 1)$

Step 2.3.1.2.3.5

Rewrite the expression.

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{- 15}{8} \cdot - 1$

Step 2.3.1.2.4

Combine $\frac{- 15}{8}$ and $- 1$ .

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{- 15 \cdot - 1}{8}$

Step 2.3.1.2.5

Multiply $- 15$ by $- 1$ .

$y - \frac{3}{16} = - \frac{x \cdot 15}{32} + \frac{15}{8}$

Step 2.3.1.3

Move $15$ to the left of $x$ .

$y - \frac{3}{16} = - \frac{15 x}{32} + \frac{15}{8}$

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 $\frac{3}{16}$ to both sides of the equation.

$y = - \frac{15 x}{32} + \frac{15}{8} + \frac{3}{16}$

Step 2.3.2.2

To write $\frac{15}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{15 x}{32} + \frac{15}{8} \cdot \frac{2}{2} + \frac{3}{16}$

Step 2.3.2.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{15}{8}$ by $\frac{2}{2}$ .

$y = - \frac{15 x}{32} + \frac{15 \cdot 2}{8 \cdot 2} + \frac{3}{16}$

Step 2.3.2.3.2

Multiply $8$ by $2$ .

$y = - \frac{15 x}{32} + \frac{15 \cdot 2}{16} + \frac{3}{16}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{15 x}{32} + \frac{15 \cdot 2 + 3}{16}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $15$ by $2$ .

$y = - \frac{15 x}{32} + \frac{30 + 3}{16}$

Step 2.3.2.5.2

Add $30$ and $3$ .

$y = - \frac{15 x}{32} + \frac{33}{16}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{15}{32} x) + \frac{33}{16}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{15}{32} x + \frac{33}{16}$

Step 3