Calculus Examples

$y = x^{3} - 2 x + 2$ , $(3, 23)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 23$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} - 2 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [2]$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 2$ by $1$ .

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

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 x^{2} - 2 + 0$

Step 1.3.2

Add $3 x^{2} - 2$ and $0$ .

$3 x^{2} - 2$

Step 1.4

Evaluate the derivative at $x = 3$ .

$3 {(3)}^{2} - 2$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 1.5.1.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 1.5.1.1.1.1

Raise $3$ to the power of $1$ .

$3^{1} {(3)}^{2} - 2$

Step 1.5.1.1.1.2

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

$3^{1 + 2} - 2$

Step 1.5.1.1.2

Add $1$ and $2$ .

$3^{3} - 2$

Step 1.5.1.2

Raise $3$ to the power of $3$ .

$27 - 2$

Step 1.5.2

Subtract $2$ from $27$ .

$25$

Step 2

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

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Step 2.1

Use the slope $25$ and a given point $(3, 23)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (23) = 25 \cdot (x - (3))$

Step 2.2

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

$y - 23 = 25 \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $25 \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 23 = 0 + 0 + 25 \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 23 = 25 \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 23 = 25 x + 25 \cdot - 3$

Step 2.3.1.4

Multiply $25$ by $- 3$ .

$y - 23 = 25 x - 75$

Step 2.3.2

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

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Step 2.3.2.1

Add $23$ to both sides of the equation.

$y = 25 x - 75 + 23$

Step 2.3.2.2

Add $- 75$ and $23$ .

$y = 25 x - 52$

Step 3