Calculus Examples

$f (x) = (x^{2} + 7) (5 x + 4)$ ; $x = 2$

Step 1

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

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Step 1.1

Substitute $2$ in for $x$ .

$y = ({(2)}^{2} + 7) (5 (2) + 4)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = (2^{2} + 7) (5 (2) + 4)$

Step 1.2.2

Remove parentheses.

$y = ({(2)}^{2} + 7) (5 (2) + 4)$

Step 1.2.3

Simplify $({(2)}^{2} + 7) (5 (2) + 4)$ .

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Step 1.2.3.1

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

$y = (4 + 7) (5 (2) + 4)$

Step 1.2.3.2

Add $4$ and $7$ .

$y = 11 (5 (2) + 4)$

Step 1.2.3.3

Multiply $5$ by $2$ .

$y = 11 (10 + 4)$

Step 1.2.3.4

Add $10$ and $4$ .

$y = 11 \cdot 14$

Step 1.2.3.5

Multiply $11$ by $14$ .

$y = 154$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 154$ 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 $f (x) = x^{2} + 7$ and $g (x) = 5 x + 4$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $5$ by $1$ .

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

Step 2.2.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Add $5$ and $0$ .

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

Step 2.2.6.2

Move $5$ to the left of $x^{2} + 7$ .

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

Step 2.2.7

By the Sum Rule, the derivative of $x^{2} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7]$ .

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

Step 2.2.8

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

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

Step 2.2.9

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$5 (x^{2} + 7) + (5 x + 4) (2 x + 0)$

Step 2.2.10

Simplify the expression.

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Step 2.2.10.1

Add $2 x$ and $0$ .

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

Step 2.2.10.2

Move $2$ to the left of $5 x + 4$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.3.4

Combine terms.

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Step 2.3.4.1

Multiply $5$ by $7$ .

$5 x^{2} + 35 + 2 (5 x) x + 2 \cdot 4 x$

Step 2.3.4.2

Multiply $5$ by $2$ .

$5 x^{2} + 35 + 10 x \cdot x + 2 \cdot 4 x$

Step 2.3.4.3

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

$5 x^{2} + 35 + 10 (x^{1} x) + 2 \cdot 4 x$

Step 2.3.4.4

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

$5 x^{2} + 35 + 10 (x^{1} x^{1}) + 2 \cdot 4 x$

Step 2.3.4.5

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

$5 x^{2} + 35 + 10 x^{1 + 1} + 2 \cdot 4 x$

Step 2.3.4.6

Add $1$ and $1$ .

$5 x^{2} + 35 + 10 x^{2} + 2 \cdot 4 x$

Step 2.3.4.7

Multiply $2$ by $4$ .

$5 x^{2} + 35 + 10 x^{2} + 8 x$

Step 2.3.4.8

Add $5 x^{2}$ and $10 x^{2}$ .

$15 x^{2} + 35 + 8 x$

Step 2.3.5

Reorder terms.

$15 x^{2} + 8 x + 35$

Step 2.4

Evaluate the derivative at $x = 2$ .

$15 {(2)}^{2} + 8 (2) + 35$

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

$15 \cdot 4 + 8 (2) + 35$

Step 2.5.1.2

Multiply $15$ by $4$ .

$60 + 8 (2) + 35$

Step 2.5.1.3

Multiply $8$ by $2$ .

$60 + 16 + 35$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add $60$ and $16$ .

$76 + 35$

Step 2.5.2.2

Add $76$ and $35$ .

$111$

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 $111$ and a given point $(2, 154)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (154) = 111 \cdot (x - (2))$

Step 3.2

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

$y - 154 = 111 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $111 \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - 154 = 0 + 0 + 111 \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 154 = 111 \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 154 = 111 x + 111 \cdot - 2$

Step 3.3.1.4

Multiply $111$ by $- 2$ .

$y - 154 = 111 x - 222$

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 $154$ to both sides of the equation.

$y = 111 x - 222 + 154$

Step 3.3.2.2

Add $- 222$ and $154$ .

$y = 111 x - 68$

Step 4