Calculus Examples

$y = \frac{3 x}{{(2 x - 5)}^{2}}$ , $(2, 6)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 6$ to find the slope of the tangent line.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 x}{{(2 x - 5)}^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{x}{{(2 x - 5)}^{2}}]$ .

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

Step 1.2

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 $g (x) = {(2 x - 5)}^{2}$ .

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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Step 1.3.1.1

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

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

Step 1.3.1.2

Multiply $2$ by $2$ .

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

Step 1.3.2

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

$3 \frac{{(2 x - 5)}^{2} \cdot 1 - x \frac{d}{d x} [{(2 x - 5)}^{2}]}{{(2 x - 5)}^{4}}$

Step 1.3.3

Multiply ${(2 x - 5)}^{2}$ by $1$ .

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

Step 1.4

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 $g (x) = 2 x - 5$ .

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Step 1.4.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

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

Step 1.4.2

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

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

Step 1.4.3

Replace all occurrences of $u$ with $2 x - 5$ .

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

Step 1.5

Simplify with factoring out.

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Step 1.5.1

Multiply $2$ by $- 1$ .

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

Step 1.5.2

Factor $2 x - 5$ out of ${(2 x - 5)}^{2} - 2 x ((2 x - 5) \frac{d}{d x} [2 x - 5])$ .

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Step 1.5.2.1

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

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

Step 1.5.2.2

Factor $2 x - 5$ out of $- 2 x ((2 x - 5) \frac{d}{d x} [2 x - 5])$ .

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

Step 1.5.2.3

Factor $2 x - 5$ out of $(2 x - 5) (2 x - 5) + (2 x - 5) (- 2 x (\frac{d}{d x} [2 x - 5]))$ .

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

Step 1.6

Cancel the common factors.

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Step 1.6.1

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

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

Step 1.6.2

Cancel the common factor.

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

Step 1.6.3

Rewrite the expression.

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

Step 1.7

By the Sum Rule, the derivative of $2 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

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

Step 1.8

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

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

Step 1.9

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

$3 \frac{2 x - 5 - 2 x (2 \cdot 1 + \frac{d}{d x} [- 5])}{{(2 x - 5)}^{3}}$

Step 1.10

Multiply $2$ by $1$ .

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

Step 1.11

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

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

Step 1.12

Simplify terms.

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Step 1.12.1

Add $2$ and $0$ .

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

Step 1.12.2

Multiply $2$ by $- 2$ .

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

Step 1.12.3

Subtract $4 x$ from $2 x$ .

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

Step 1.12.4

Combine $3$ and $\frac{- 2 x - 5}{{(2 x - 5)}^{3}}$ .

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

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

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

Step 1.13.2

Simplify each term.

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Step 1.13.2.1

Multiply $- 2$ by $3$ .

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

Step 1.13.2.2

Multiply $3$ by $- 5$ .

$\frac{- 6 x - 15}{{(2 x - 5)}^{3}}$

Step 1.13.3

Factor $3$ out of $- 6 x - 15$ .

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Step 1.13.3.1

Factor $3$ out of $- 6 x$ .

$\frac{3 (- 2 x) - 15}{{(2 x - 5)}^{3}}$

Step 1.13.3.2

Factor $3$ out of $- 15$ .

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

Step 1.13.3.3

Factor $3$ out of $3 (- 2 x) + 3 (- 5)$ .

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

Step 1.13.4

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

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

Step 1.13.5

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

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

Step 1.13.6

Factor $- 1$ out of $- (2 x) - 1 (5)$ .

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

Step 1.13.7

Rewrite $- (2 x + 5)$ as $- 1 (2 x + 5)$ .

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

Step 1.13.8

Move the negative in front of the fraction.

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

Step 1.14

Evaluate the derivative at $x = 2$ .

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

Step 1.15

Simplify.

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Step 1.15.1

Simplify the numerator.

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Step 1.15.1.1

Multiply $2$ by $2$ .

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

Step 1.15.1.2

Add $4$ and $5$ .

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

Step 1.15.2

Simplify the denominator.

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Step 1.15.2.1

Multiply $2$ by $2$ .

$- \frac{3 \cdot 9}{{(4 - 5)}^{3}}$

Step 1.15.2.2

Subtract $5$ from $4$ .

$- \frac{3 \cdot 9}{{(- 1)}^{3}}$

Step 1.15.2.3

Raise $- 1$ to the power of $3$ .

$- \frac{3 \cdot 9}{- 1}$

Step 1.15.3

Simplify the expression.

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Step 1.15.3.1

Multiply $3$ by $9$ .

$- \frac{27}{- 1}$

Step 1.15.3.2

Divide $27$ by $- 1$ .

$- - 27$

Step 1.15.3.3

Multiply $- 1$ by $- 27$ .

$27$

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 $27$ and a given point $(2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = 27 \cdot (x - (2))$

Step 2.2

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

$y - 6 = 27 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $27 \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 + 27 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 6 = 27 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 6 = 27 x + 27 \cdot - 2$

Step 2.3.1.4

Multiply $27$ by $- 2$ .

$y - 6 = 27 x - 54$

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 $6$ to both sides of the equation.

$y = 27 x - 54 + 6$

Step 2.3.2.2

Add $- 54$ and $6$ .

$y = 27 x - 48$

Step 3