Calculus Examples

$f (x) = \frac{x}{{(3 x - 5)}^{9}}$ , $x = 2$

Step 1

Find the corresponding $y$ -value to $x = 2$ .

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Step 1.1

Substitute $2$ in for $x$ .

$y = \frac{2}{{(3 (2) - 5)}^{9}}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{2}{{(3 (2) - 5)}^{9}}$

Step 1.2.2

Simplify $\frac{2}{{(3 (2) - 5)}^{9}}$ .

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Step 1.2.2.1

Simplify the denominator.

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Step 1.2.2.1.1

Multiply $3$ by $2$ .

$y = \frac{2}{{(6 - 5)}^{9}}$

Step 1.2.2.1.2

Subtract $5$ from $6$ .

$y = \frac{2}{1^{9}}$

Step 1.2.2.1.3

One to any power is one.

$y = \frac{2}{1}$

Step 1.2.2.2

Divide $2$ by $1$ .

$y = 2$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 2$ to find the slope of the tangent line.

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Step 2.1

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 $f (x) = x$ and $g (x) = {(3 x - 5)}^{9}$ .

$\frac{{(3 x - 5)}^{9} \frac{d}{d x} [x] - x \frac{d}{d x} [{(3 x - 5)}^{9}]}{{({(3 x - 5)}^{9})}^{2}}$

Step 2.2

Differentiate using the Power Rule.

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Step 2.2.1

Multiply the exponents in ${({(3 x - 5)}^{9})}^{2}$ .

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Step 2.2.1.1

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

$\frac{{(3 x - 5)}^{9} \frac{d}{d x} [x] - x \frac{d}{d x} [{(3 x - 5)}^{9}]}{{(3 x - 5)}^{9 \cdot 2}}$

Step 2.2.1.2

Multiply $9$ by $2$ .

$\frac{{(3 x - 5)}^{9} \frac{d}{d x} [x] - x \frac{d}{d x} [{(3 x - 5)}^{9}]}{{(3 x - 5)}^{18}}$

Step 2.2.2

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

$\frac{{(3 x - 5)}^{9} \cdot 1 - x \frac{d}{d x} [{(3 x - 5)}^{9}]}{{(3 x - 5)}^{18}}$

Step 2.2.3

Multiply ${(3 x - 5)}^{9}$ by $1$ .

$\frac{{(3 x - 5)}^{9} - x \frac{d}{d x} [{(3 x - 5)}^{9}]}{{(3 x - 5)}^{18}}$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = 3 x - 5$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $3 x - 5$ .

$\frac{{(3 x - 5)}^{9} - x (\frac{d}{d u} [u^{9}] \frac{d}{d x} [3 x - 5])}{{(3 x - 5)}^{18}}$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 9$ .

$\frac{{(3 x - 5)}^{9} - x (9 u^{8} \frac{d}{d x} [3 x - 5])}{{(3 x - 5)}^{18}}$

Step 2.3.3

Replace all occurrences of $u$ with $3 x - 5$ .

$\frac{{(3 x - 5)}^{9} - x (9 {(3 x - 5)}^{8} \frac{d}{d x} [3 x - 5])}{{(3 x - 5)}^{18}}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Multiply $9$ by $- 1$ .

$\frac{{(3 x - 5)}^{9} - 9 x ({(3 x - 5)}^{8} \frac{d}{d x} [3 x - 5])}{{(3 x - 5)}^{18}}$

Step 2.4.2

Factor ${(3 x - 5)}^{8}$ out of ${(3 x - 5)}^{9} - 9 x ({(3 x - 5)}^{8} \frac{d}{d x} [3 x - 5])$ .

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Step 2.4.2.1

Factor ${(3 x - 5)}^{8}$ out of ${(3 x - 5)}^{9}$ .

$\frac{{(3 x - 5)}^{8} (3 x - 5) - 9 x ({(3 x - 5)}^{8} \frac{d}{d x} [3 x - 5])}{{(3 x - 5)}^{18}}$

Step 2.4.2.2

Factor ${(3 x - 5)}^{8}$ out of $- 9 x ({(3 x - 5)}^{8} \frac{d}{d x} [3 x - 5])$ .

$\frac{{(3 x - 5)}^{8} (3 x - 5) + {(3 x - 5)}^{8} (- 9 x (\frac{d}{d x} [3 x - 5]))}{{(3 x - 5)}^{18}}$

Step 2.4.2.3

Factor ${(3 x - 5)}^{8}$ out of ${(3 x - 5)}^{8} (3 x - 5) + {(3 x - 5)}^{8} (- 9 x (\frac{d}{d x} [3 x - 5]))$ .

$\frac{{(3 x - 5)}^{8} (3 x - 5 - 9 x (\frac{d}{d x} [3 x - 5]))}{{(3 x - 5)}^{18}}$

Step 2.5

Cancel the common factors.

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Step 2.5.1

Factor ${(3 x - 5)}^{8}$ out of ${(3 x - 5)}^{18}$ .

$\frac{{(3 x - 5)}^{8} (3 x - 5 - 9 x (\frac{d}{d x} [3 x - 5]))}{{(3 x - 5)}^{8} {(3 x - 5)}^{10}}$

Step 2.5.2

Cancel the common factor.

$\frac{{(3 x - 5)}^{8} (3 x - 5 - 9 x (\frac{d}{d x} [3 x - 5]))}{{(3 x - 5)}^{8} {(3 x - 5)}^{10}}$

Step 2.5.3

Rewrite the expression.

$\frac{3 x - 5 - 9 x (\frac{d}{d x} [3 x - 5])}{{(3 x - 5)}^{10}}$

Step 2.6

By the Sum Rule, the derivative of $3 x - 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

$\frac{3 x - 5 - 9 x (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5])}{{(3 x - 5)}^{10}}$

Step 2.7

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

$\frac{3 x - 5 - 9 x (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])}{{(3 x - 5)}^{10}}$

Step 2.8

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

$\frac{3 x - 5 - 9 x (3 \cdot 1 + \frac{d}{d x} [- 5])}{{(3 x - 5)}^{10}}$

Step 2.9

Multiply $3$ by $1$ .

$\frac{3 x - 5 - 9 x (3 + \frac{d}{d x} [- 5])}{{(3 x - 5)}^{10}}$

Step 2.10

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{3 x - 5 - 9 x (3 + 0)}{{(3 x - 5)}^{10}}$

Step 2.11

Simplify by adding terms.

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Step 2.11.1

Add $3$ and $0$ .

$\frac{3 x - 5 - 9 x \cdot 3}{{(3 x - 5)}^{10}}$

Step 2.11.2

Multiply $3$ by $- 9$ .

$\frac{3 x - 5 - 27 x}{{(3 x - 5)}^{10}}$

Step 2.11.3

Subtract $27 x$ from $3 x$ .

$\frac{- 24 x - 5}{{(3 x - 5)}^{10}}$

Step 2.12

Simplify.

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Step 2.12.1

Factor $- 1$ out of $- 24 x$ .

$\frac{- (24 x) - 5}{{(3 x - 5)}^{10}}$

Step 2.12.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (24 x) - 1 (5)}{{(3 x - 5)}^{10}}$

Step 2.12.3

Factor $- 1$ out of $- (24 x) - 1 (5)$ .

$\frac{- (24 x + 5)}{{(3 x - 5)}^{10}}$

Step 2.12.4

Rewrite $- (24 x + 5)$ as $- 1 (24 x + 5)$ .

$\frac{- 1 (24 x + 5)}{{(3 x - 5)}^{10}}$

Step 2.12.5

Move the negative in front of the fraction.

$- \frac{24 x + 5}{{(3 x - 5)}^{10}}$

Step 2.13

Evaluate the derivative at $x = 2$ .

$- \frac{24 (2) + 5}{{(3 (2) - 5)}^{10}}$

Step 2.14

Simplify.

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Step 2.14.1

Simplify the numerator.

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Step 2.14.1.1

Multiply $24$ by $2$ .

$- \frac{48 + 5}{{(3 (2) - 5)}^{10}}$

Step 2.14.1.2

Add $48$ and $5$ .

$- \frac{53}{{(3 (2) - 5)}^{10}}$

Step 2.14.2

Simplify the denominator.

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Step 2.14.2.1

Multiply $3$ by $2$ .

$- \frac{53}{{(6 - 5)}^{10}}$

Step 2.14.2.2

Subtract $5$ from $6$ .

$- \frac{53}{1^{10}}$

Step 2.14.2.3

One to any power is one.

$- \frac{53}{1}$

Step 2.14.3

Simplify the expression.

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Step 2.14.3.1

Divide $53$ by $1$ .

$- 1 \cdot 53$

Step 2.14.3.2

Multiply $- 1$ by $53$ .

$- 53$

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 $- 53$ and a given point $(2, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - 53 \cdot (x - (2))$

Step 3.2

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

$y - 2 = - 53 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 53 \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - 2 = 0 + 0 - 53 \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 2 = - 53 \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 2 = - 53 x - 53 \cdot - 2$

Step 3.3.1.4

Multiply $- 53$ by $- 2$ .

$y - 2 = - 53 x + 106$

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 $2$ to both sides of the equation.

$y = - 53 x + 106 + 2$

Step 3.3.2.2

Add $106$ and $2$ .

$y = - 53 x + 108$

Step 4