Calculus Examples

$y = {(x + 3)}^{2}$ , $(1, 16)$

Find $\frac{d y}{d x}$ and evaluate at $x = 1$ and $y = 16$ to find the slope of the tangent line at $x = 1$ and $y = 16$ .

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Simplify ${(x + 3)}^{2}$ .

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Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$y = (x + 3) (x + 3)$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Apply the distributive property.

$y = x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$y = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$y = x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$y = x^{2} + 6 x + 9$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} + 6 x + 9)$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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Differentiate.

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By the Sum Rule, the derivative of $x^{2} + 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [9]$

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

$2 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [9]$

Evaluate $\frac{d}{d x} [6 x]$ .

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Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$2 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

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

$2 x + 6 \cdot 1 + \frac{d}{d x} [9]$

Multiply $6$ by $1$ .

$2 x + 6 + \frac{d}{d x} [9]$

Differentiate using the Constant Rule.

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Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x + 6 + 0$

Add $2 x + 6$ and $0$ .

$2 x + 6$

Reform the equation by setting the left side equal to the right side.

$y' = 2 x + 6$

Replace $y'$ with $\frac{d y}{d x}$ .

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

Evaluate at $x = 1$ and $y = 16$ .

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Replace the variable $x$ with $1$ in the expression.

$2 (1) + 6$

Multiply $2$ by $1$ .

$2 + 6$

Add $2$ and $6$ .

$8$

Plug in the slope of the tangent line and the $x$ and $y$ values of the point into the point-slope formula $y - y_{1} = m (x - x_{1})$ .

$y - (16) = 8 \cdot (x - (1))$

Simplify.

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The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Rewrite in slope-intercept form.

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Multiply $- 1$ by $16$ .

$y - 16 = 8 \cdot (x - (1))$

Simplify $8 \cdot (x - (1))$ .

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Multiply $- 1$ by $1$ .

$y - 16 = 8 \cdot (x - 1)$

Apply the distributive property.

$y - 16 = 8 x + 8 \cdot - 1$

Multiply $8$ by $- 1$ .

$y - 16 = 8 x - 8$

Move all terms not containing $y$ to the right side of the equation.

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Add $16$ to both sides of the equation.

$y = 8 x - 8 + 16$

Add $- 8$ and $16$ .

$y = 8 x + 8$

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