Calculus Examples

$y = 4 x^{2} + 4$ , $(- 3, 40)$

Find $\frac{d y}{d x}$ and evaluate at $x = - 3$ and $y = 40$ to find the slope of the tangent line at $x = - 3$ and $y = 40$ .

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Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (4 x^{2} + 4)$

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

$y'$

Differentiate the right side of the equation.

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

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [4]$

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

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

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

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [4]$

Differentiate using the Constant Rule.

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

$8 x + 0$

Add $8 x$ and $0$ .

$8 x$

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

$y' = 8 x$

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

$\frac{d y}{d x} = 8 x$

Evaluate at $x = - 3$ and $y = 40$ .

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Replace the variable $x$ with $- 3$ in the expression.

$8 (- 3)$

Multiply $8$ by $- 3$ .

$- 24$

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

$y - (40) = - 24 \cdot (x - (- 3))$

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 $40$ .

$y - 40 = - 24 \cdot (x - (- 3))$

Simplify $- 24 \cdot (x - (- 3))$ .

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Multiply $- 1$ by $- 3$ .

$y - 40 = - 24 \cdot (x + 3)$

Apply the distributive property.

$y - 40 = - 24 x - 24 \cdot 3$

Multiply $- 24$ by $3$ .

$y - 40 = - 24 x - 72$

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

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Add $40$ to both sides of the equation.

$y = - 24 x - 72 + 40$

Add $- 72$ and $40$ .

$y = - 24 x - 32$

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