Calculus Examples

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

Find the first derivative and evaluate at $x = 1$ and $y = 16$ to find the slope of the tangent line.

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

$\frac{d}{d x} [(x + 3) (x + 3)]$

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

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Apply the distributive property.

$\frac{d}{d x} [x (x + 3) + 3 (x + 3)]$

Apply the distributive property.

$\frac{d}{d x} [x \cdot x + x \cdot 3 + 3 (x + 3)]$

Apply the distributive property.

$\frac{d}{d x} [x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3]$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{2} + x \cdot 3 + 3 x + 3 \cdot 3]$

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

$\frac{d}{d x} [x^{2} + 3 \cdot x + 3 x + 3 \cdot 3]$

Multiply $3$ by $3$ .

$\frac{d}{d x} [x^{2} + 3 x + 3 x + 9]$

Add $3 x$ and $3 x$ .

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

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 $n = 2$ .

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

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 $1$ .

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

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 $0$ .

$2 x + 6$

Evaluate the derivative at $x = 1$ .

$2 (1) + 6$

Simplify.

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Multiply $2$ by $1$ .

$2 + 6$

Add $2$ and $6$ .

$8$

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Use the slope $8$ and a given point $(1, 16)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Solve for $y$ .

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

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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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