Calculus Examples

$y = 3 x^{3} + 4 x + 5$ , $(2, 37)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 37$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $3 x^{3} + 4 x + 5$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [5]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [5]$

Step 1.2.2

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

$3 (3 x^{2}) + \frac{d}{d x} [4 x] + \frac{d}{d x} [5]$

Step 1.2.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [5]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$9 x^{2} + 4 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.3.2

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

$9 x^{2} + 4 \cdot 1 + \frac{d}{d x} [5]$

Step 1.3.3

Multiply $4$ by $1$ .

$9 x^{2} + 4 + \frac{d}{d x} [5]$

Step 1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$9 x^{2} + 4 + 0$

Step 1.4.2

Add $9 x^{2} + 4$ and $0$ .

$9 x^{2} + 4$

Step 1.5

Evaluate the derivative at $x = 2$ .

$9 {(2)}^{2} + 4$

Step 1.6

Simplify.

Tap for more steps...

Step 1.6.1

Simplify each term.

Tap for more steps...

Step 1.6.1.1

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

$9 \cdot 4 + 4$

Step 1.6.1.2

Multiply $9$ by $4$ .

$36 + 4$

Step 1.6.2

Add $36$ and $4$ .

$40$

Step 2

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

Tap for more steps...

Step 2.1

Use the slope $40$ and a given point $(2, 37)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (37) = 40 \cdot (x - (2))$

Step 2.2

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

$y - 37 = 40 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $40 \cdot (x - 2)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 37 = 0 + 0 + 40 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 37 = 40 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 37 = 40 x + 40 \cdot - 2$

Step 2.3.1.4

Multiply $40$ by $- 2$ .

$y - 37 = 40 x - 80$

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

Add $37$ to both sides of the equation.

$y = 40 x - 80 + 37$

Step 2.3.2.2

Add $- 80$ and $37$ .

$y = 40 x - 43$

Step 3

Enter YOUR Problem