Calculus Examples

$y = \frac{x^{4}}{{(x^{2} - 6)}^{5}}$ , $(- 2, - \frac{1}{2})$

Step 1

Find the first derivative and evaluate at $x = - 2$ and $y = - \frac{1}{2}$ to find the slope of the tangent line.

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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 $f (x) = x^{4}$ and $g (x) = {(x^{2} - 6)}^{5}$ .

$\frac{{(x^{2} - 6)}^{5} \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [{(x^{2} - 6)}^{5}]}{{({(x^{2} - 6)}^{5})}^{2}}$

Step 1.2

Differentiate using the Power Rule.

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Step 1.2.1

Multiply the exponents in ${({(x^{2} - 6)}^{5})}^{2}$ .

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Step 1.2.1.1

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

$\frac{{(x^{2} - 6)}^{5} \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [{(x^{2} - 6)}^{5}]}{{(x^{2} - 6)}^{5 \cdot 2}}$

Step 1.2.1.2

Multiply $5$ by $2$ .

$\frac{{(x^{2} - 6)}^{5} \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [{(x^{2} - 6)}^{5}]}{{(x^{2} - 6)}^{10}}$

Step 1.2.2

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

$\frac{{(x^{2} - 6)}^{5} (4 x^{3}) - x^{4} \frac{d}{d x} [{(x^{2} - 6)}^{5}]}{{(x^{2} - 6)}^{10}}$

Step 1.2.3

Move $4$ to the left of ${(x^{2} - 6)}^{5}$ .

$\frac{4 \cdot {(x^{2} - 6)}^{5} x^{3} - x^{4} \frac{d}{d x} [{(x^{2} - 6)}^{5}]}{{(x^{2} - 6)}^{10}}$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{2} - 6$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 6$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - x^{4} (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{10}}$

Step 1.3.2

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

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - x^{4} (5 u^{4} \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{10}}$

Step 1.3.3

Replace all occurrences of $u$ with $x^{2} - 6$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - x^{4} (5 {(x^{2} - 6)}^{4} \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{10}}$

Step 1.4

Differentiate.

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Step 1.4.1

Multiply $5$ by $- 1$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 5 x^{4} ({(x^{2} - 6)}^{4} \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{10}}$

Step 1.4.2

By the Sum Rule, the derivative of $x^{2} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 5 x^{4} ({(x^{2} - 6)}^{4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]))}{{(x^{2} - 6)}^{10}}$

Step 1.4.3

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

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 5 x^{4} ({(x^{2} - 6)}^{4} (2 x + \frac{d}{d x} [- 6]))}{{(x^{2} - 6)}^{10}}$

Step 1.4.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 5 x^{4} ({(x^{2} - 6)}^{4} (2 x + 0))}{{(x^{2} - 6)}^{10}}$

Step 1.4.5

Simplify the expression.

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Step 1.4.5.1

Add $2 x$ and $0$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 5 x^{4} ({(x^{2} - 6)}^{4} (2 x))}{{(x^{2} - 6)}^{10}}$

Step 1.4.5.2

Move $2$ to the left of ${(x^{2} - 6)}^{4}$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 5 x^{4} (2 \cdot {(x^{2} - 6)}^{4} x)}{{(x^{2} - 6)}^{10}}$

Step 1.4.5.3

Multiply $2$ by $- 5$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 10 x^{4} ({(x^{2} - 6)}^{4} x)}{{(x^{2} - 6)}^{10}}$

Step 1.5

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

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 10 (x^{1} x^{4}) {(x^{2} - 6)}^{4}}{{(x^{2} - 6)}^{10}}$

Step 1.6

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

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 10 x^{1 + 4} {(x^{2} - 6)}^{4}}{{(x^{2} - 6)}^{10}}$

Step 1.7

Add $1$ and $4$ .

$\frac{4 {(x^{2} - 6)}^{5} x^{3} - 10 x^{5} {(x^{2} - 6)}^{4}}{{(x^{2} - 6)}^{10}}$

Step 1.8

Simplify.

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Step 1.8.1

Simplify the numerator.

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Step 1.8.1.1

Factor $2 {(x^{2} - 6)}^{4} x^{3}$ out of $4 {(x^{2} - 6)}^{5} x^{3} - 10 x^{5} {(x^{2} - 6)}^{4}$ .

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Step 1.8.1.1.1

Factor $2 {(x^{2} - 6)}^{4} x^{3}$ out of $4 {(x^{2} - 6)}^{5} x^{3}$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (2 (x^{2} - 6)) - 10 x^{5} {(x^{2} - 6)}^{4}}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.1.2

Factor $2 {(x^{2} - 6)}^{4} x^{3}$ out of $- 10 x^{5} {(x^{2} - 6)}^{4}$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (2 (x^{2} - 6)) + 2 {(x^{2} - 6)}^{4} x^{3} (- 5 x^{2})}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.1.3

Factor $2 {(x^{2} - 6)}^{4} x^{3}$ out of $2 {(x^{2} - 6)}^{4} x^{3} (2 (x^{2} - 6)) + 2 {(x^{2} - 6)}^{4} x^{3} (- 5 x^{2})$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (2 (x^{2} - 6) - 5 x^{2})}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.2

Apply the distributive property.

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (2 x^{2} + 2 \cdot - 6 - 5 x^{2})}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.3

Multiply $2$ by $- 6$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (2 x^{2} - 12 - 5 x^{2})}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.4

Subtract $5 x^{2}$ from $2 x^{2}$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (- 3 x^{2} - 12)}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.5

Factor $3$ out of $- 3 x^{2} - 12$ .

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Step 1.8.1.5.1

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

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (3 (- x^{2}) - 12)}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.5.2

Factor $3$ out of $- 12$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (3 (- x^{2}) + 3 (- 4))}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.5.3

Factor $3$ out of $3 (- x^{2}) + 3 (- 4)$ .

$\frac{2 {(x^{2} - 6)}^{4} x^{3} (3 (- x^{2} - 4))}{{(x^{2} - 6)}^{10}}$

$\frac{2 {(x^{2} - 6)}^{4} x^{3} \cdot 3 (- x^{2} - 4)}{{(x^{2} - 6)}^{10}}$

Step 1.8.1.6

Multiply $3$ by $2$ .

$\frac{6 {(x^{2} - 6)}^{4} x^{3} (- x^{2} - 4)}{{(x^{2} - 6)}^{10}}$

Step 1.8.2

Cancel the common factor of ${(x^{2} - 6)}^{4}$ and ${(x^{2} - 6)}^{10}$ .

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Step 1.8.2.1

Factor ${(x^{2} - 6)}^{4}$ out of $6 {(x^{2} - 6)}^{4} x^{3} (- x^{2} - 4)$ .

$\frac{{(x^{2} - 6)}^{4} (6 x^{3} (- x^{2} - 4))}{{(x^{2} - 6)}^{10}}$

Step 1.8.2.2

Cancel the common factors.

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Step 1.8.2.2.1

Factor ${(x^{2} - 6)}^{4}$ out of ${(x^{2} - 6)}^{10}$ .

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

Step 1.8.2.2.2

Cancel the common factor.

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

Step 1.8.2.2.3

Rewrite the expression.

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

Step 1.8.3

Factor $- 1$ out of $- x^{2}$ .

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

Step 1.8.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 1.8.5

Factor $- 1$ out of $- (x^{2}) - 1 (4)$ .

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

Step 1.8.6

Rewrite $- (x^{2} + 4)$ as $- 1 (x^{2} + 4)$ .

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

Step 1.8.7

Move the negative in front of the fraction.

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

Step 1.9

Evaluate the derivative at $x = - 2$ .

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

Step 1.10

Simplify.

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Step 1.10.1

Simplify the numerator.

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Step 1.10.1.1

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

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

Step 1.10.1.2

Add $4$ and $4$ .

$- \frac{6 {(- 2)}^{3} \cdot 8}{{({(- 2)}^{2} - 6)}^{6}}$

Step 1.10.1.3

Multiply $8$ by $6$ .

$- \frac{48 {(- 2)}^{3}}{{({(- 2)}^{2} - 6)}^{6}}$

Step 1.10.1.4

Raise $- 2$ to the power of $3$ .

$- \frac{48 \cdot - 8}{{({(- 2)}^{2} - 6)}^{6}}$

Step 1.10.2

Simplify the denominator.

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Step 1.10.2.1

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

$- \frac{48 \cdot - 8}{{(4 - 6)}^{6}}$

Step 1.10.2.2

Subtract $6$ from $4$ .

$- \frac{48 \cdot - 8}{{(- 2)}^{6}}$

Step 1.10.2.3

Raise $- 2$ to the power of $6$ .

$- \frac{48 \cdot - 8}{64}$

Step 1.10.3

Simplify the expression.

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Step 1.10.3.1

Multiply $48$ by $- 8$ .

$- \frac{- 384}{64}$

Step 1.10.3.2

Divide $- 384$ by $64$ .

$- - 6$

Step 1.10.3.3

Multiply $- 1$ by $- 6$ .

$6$

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 $6$ and a given point $(- 2, - \frac{1}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{1}{2}) = 6 \cdot (x - (- 2))$

Step 2.2

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

$y + \frac{1}{2} = 6 \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $6 \cdot (x + 2)$ .

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Step 2.3.1.1

Rewrite.

$y + \frac{1}{2} = 0 + 0 + 6 \cdot (x + 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y + \frac{1}{2} = 6 \cdot (x + 2)$

Step 2.3.1.3

Apply the distributive property.

$y + \frac{1}{2} = 6 x + 6 \cdot 2$

Step 2.3.1.4

Multiply $6$ by $2$ .

$y + \frac{1}{2} = 6 x + 12$

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

Subtract $\frac{1}{2}$ from both sides of the equation.

$y = 6 x + 12 - \frac{1}{2}$

Step 2.3.2.2

To write $12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 6 x + 12 \cdot \frac{2}{2} - \frac{1}{2}$

Step 2.3.2.3

Combine $12$ and $\frac{2}{2}$ .

$y = 6 x + \frac{12 \cdot 2}{2} - \frac{1}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = 6 x + \frac{12 \cdot 2 - 1}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $12$ by $2$ .

$y = 6 x + \frac{24 - 1}{2}$

Step 2.3.2.5.2

Subtract $1$ from $24$ .

$y = 6 x + \frac{23}{2}$

Step 3