Calculus Examples

$y = 3 x^{3} + x + 3$ , $(5, 383)$

Find $\frac{d y}{d x}$ and evaluate at $x = 5$ and $y = 383$ to find the slope of the tangent line at $x = 5$ and $y = 383$ .

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Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3 x^{3} + x + 3)$

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 $3 x^{3} + x + 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

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

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Multiply $3$ by $3$ .

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

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

$9 x^{2} + 1 + \frac{d}{d x} [3]$

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$9 x^{2} + 1 + 0$

Add $9 x^{2} + 1$ and $0$ .

$9 x^{2} + 1$

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

$y' = 9 x^{2} + 1$

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

$\frac{d y}{d x} = 9 x^{2} + 1$

Evaluate at $x = 5$ .

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Replace the variable $x$ with $5$ in the expression.

$9 {(5)}^{2} + 1$

Simplify each term.

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Remove parentheses around $5$ .

$9 \cdot 5^{2} + 1$

Raise $5$ to the power of $2$ .

$9 \cdot 25 + 1$

Multiply $9$ by $25$ .

$225 + 1$

Add $225$ and $1$ .

$226$

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

$y - (383) = 226 \cdot (x - (5))$

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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Simplify the right side.

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Simplify the expression.

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Multiply $- 1$ by $5$ .

$y - (383) = 226 \cdot (x - 5)$

Multiply $226$ by $x - 5$ .

$y - (383) = 226 (x - 5)$

Apply the distributive property.

$y - (383) = 226 x + 226 \cdot - 5$

Multiply $226$ by $- 5$ .

$y - (383) = 226 x - 1130$

Multiply $- 1$ by $383$ .

$y - 383 = 226 x - 1130$

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

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Add $383$ to both sides of the equation.

$y = 383 + 226 x - 1130$

Subtract $1130$ from $383$ .

$y = 226 x - 747$

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