Calculus Examples

$f (x) = x {(x^{2} - 4 x + 5)}^{8}$ ; $x = 2$

Step 1

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

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Step 1.1

Substitute $2$ in for $x$ .

$y = (2) {({(2)}^{2} - 4 \cdot 2 + 5)}^{8}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2 {(2^{2} - 4 \cdot 2 + 5)}^{8}$

Step 1.2.2

Remove parentheses.

$y = (2) {({(2)}^{2} - 4 \cdot 2 + 5)}^{8}$

Step 1.2.3

Simplify $(2) {({(2)}^{2} - 4 \cdot 2 + 5)}^{8}$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

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

$y = 2 {(4 - 4 \cdot 2 + 5)}^{8}$

Step 1.2.3.1.2

Multiply $- 4$ by $2$ .

$y = 2 {(4 - 8 + 5)}^{8}$

Step 1.2.3.2

Simplify the expression.

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Step 1.2.3.2.1

Subtract $8$ from $4$ .

$y = 2 {(- 4 + 5)}^{8}$

Step 1.2.3.2.2

Add $- 4$ and $5$ .

$y = 2 \cdot 1^{8}$

Step 1.2.3.2.3

One to any power is one.

$y = 2 \cdot 1$

Step 1.2.3.2.4

Multiply $2$ 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 Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x^{2} - 4 x + 5)}^{8}$ .

$x \frac{d}{d x} [{(x^{2} - 4 x + 5)}^{8}] + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{8}$ and $g (x) = x^{2} - 4 x + 5$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 4 x + 5$ .

$x (\frac{d}{d u} [u^{8}] \frac{d}{d x} [x^{2} - 4 x + 5]) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$x (8 u^{7} \frac{d}{d x} [x^{2} - 4 x + 5]) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} - 4 x + 5$ .

$x (8 {(x^{2} - 4 x + 5)}^{7} \frac{d}{d x} [x^{2} - 4 x + 5]) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $x^{2} - 4 x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$ .

$x (8 {(x^{2} - 4 x + 5)}^{7} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5])) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.2

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

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5])) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [5])) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.4

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

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4 \cdot 1 + \frac{d}{d x} [5])) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.5

Multiply $- 4$ by $1$ .

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4 + \frac{d}{d x} [5])) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.6

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

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4 + 0)) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.7

Add $2 x - 4$ and $0$ .

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4)) + {(x^{2} - 4 x + 5)}^{8} \frac{d}{d x} [x]$

Step 2.3.8

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

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4)) + {(x^{2} - 4 x + 5)}^{8} \cdot 1$

Step 2.3.9

Multiply ${(x^{2} - 4 x + 5)}^{8}$ by $1$ .

$x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4)) + {(x^{2} - 4 x + 5)}^{8}$

Step 2.4

Factor ${(x^{2} - 4 x + 5)}^{7}$ out of $x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4)) + {(x^{2} - 4 x + 5)}^{8}$ .

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Step 2.4.1

Factor ${(x^{2} - 4 x + 5)}^{7}$ out of $x (8 {(x^{2} - 4 x + 5)}^{7} (2 x - 4))$ .

${(x^{2} - 4 x + 5)}^{7} (x (8 (2 x - 4))) + {(x^{2} - 4 x + 5)}^{8}$

Step 2.4.2

Factor ${(x^{2} - 4 x + 5)}^{7}$ out of ${(x^{2} - 4 x + 5)}^{8}$ .

${(x^{2} - 4 x + 5)}^{7} (x (8 (2 x - 4))) + {(x^{2} - 4 x + 5)}^{7} (x^{2} - 4 x + 5)$

Step 2.4.3

Factor ${(x^{2} - 4 x + 5)}^{7}$ out of ${(x^{2} - 4 x + 5)}^{7} (x (8 (2 x - 4))) + {(x^{2} - 4 x + 5)}^{7} (x^{2} - 4 x + 5)$ .

${(x^{2} - 4 x + 5)}^{7} (x (8 (2 x - 4)) + x^{2} - 4 x + 5)$

Step 2.5

Evaluate the derivative at $x = 2$ .

${({(2)}^{2} - 4 \cdot 2 + 5)}^{7} ((2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5)$

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

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

${(4 - 4 \cdot 2 + 5)}^{7} ((2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5)$

Step 2.6.1.2

Multiply $- 4$ by $2$ .

${(4 - 8 + 5)}^{7} ((2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5)$

Step 2.6.2

Simplify the expression.

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Step 2.6.2.1

Subtract $8$ from $4$ .

${(- 4 + 5)}^{7} ((2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5)$

Step 2.6.2.2

Add $- 4$ and $5$ .

$1^{7} ((2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5)$

Step 2.6.2.3

One to any power is one.

$1 ((2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5)$

Step 2.6.2.4

Multiply $(2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5$ by $1$ .

$(2) (8 (2 (2) - 4)) + {(2)}^{2} - 4 \cdot 2 + 5$

Step 2.6.3

Simplify each term.

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Step 2.6.3.1

Multiply $2$ by $2$ .

$2 (8 (4 - 4)) + {(2)}^{2} - 4 \cdot 2 + 5$

Step 2.6.3.2

Subtract $4$ from $4$ .

$2 (8 \cdot 0) + {(2)}^{2} - 4 \cdot 2 + 5$

Step 2.6.3.3

Multiply $2 (8 \cdot 0)$ .

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Step 2.6.3.3.1

Multiply $8$ by $0$ .

$2 \cdot 0 + {(2)}^{2} - 4 \cdot 2 + 5$

Step 2.6.3.3.2

Multiply $2$ by $0$ .

$0 + {(2)}^{2} - 4 \cdot 2 + 5$

Step 2.6.3.4

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

$0 + 4 - 4 \cdot 2 + 5$

Step 2.6.3.5

Multiply $- 4$ by $2$ .

$0 + 4 - 8 + 5$

Step 2.6.4

Simplify by adding and subtracting.

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Step 2.6.4.1

Add $0$ and $4$ .

$4 - 8 + 5$

Step 2.6.4.2

Subtract $8$ from $4$ .

$- 4 + 5$

Step 2.6.4.3

Add $- 4$ and $5$ .

$1$

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 $1$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 1 \cdot (x - (2))$

Step 3.2

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

$y - 2 = 1 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Multiply $x - 2$ by $1$ .

$y - 2 = x - 2$

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.

$y = x - 2 + 2$

Step 3.3.2.2

Combine the opposite terms in $x - 2 + 2$ .

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Step 3.3.2.2.1

Add $- 2$ and $2$ .

$y = x + 0$

Step 3.3.2.2.2

Add $x$ and $0$ .

$y = x$

Step 4