Calculus Examples

$y = x^{2} + 3 x + 34$ , $(4, 62)$

Find $\frac{d y}{d x}$ and evaluate at $x = 4$ and $y = 62$ to find the slope of the tangent line at $x = 4$ and $y = 62$ .

Tap for more steps...

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} + 3 x + 34)$

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

$y'$

Differentiate the right side of the equation.

Tap for more steps...

By the Sum Rule, the derivative of $x^{2} + 3 x + 34$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

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

$2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

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

Tap for more steps...

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

$2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [34]$

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

$2 x + 3 \cdot 1 + \frac{d}{d x} [34]$

Multiply $3$ by $1$ .

$2 x + 3 + \frac{d}{d x} [34]$

Since $34$ is constant with respect to $x$ , the derivative of $34$ with respect to $x$ is $0$ .

$2 x + 3 + 0$

Add $2 x + 3$ and $0$ .

$2 x + 3$

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

$y' = 2 x + 3$

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

$\frac{d y}{d x} = 2 x + 3$

Evaluate at $x = 4$ .

Tap for more steps...

Replace the variable $x$ with $4$ in the expression.

$2 (4) + 3$

Multiply $2$ by $4$ .

$8 + 3$

Add $8$ and $3$ .

$11$

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

$y - (62) = 11 \cdot (x - (4))$

Simplify.

Tap for more steps...

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.

Tap for more steps...

Multiply $- 1$ by $62$ .

$y - 62 = 11 \cdot (x - (4))$

Simplify $11 \cdot (x - (4))$ .

Tap for more steps...

Multiply $- 1$ by $4$ .

$y - 62 = 11 \cdot (x - 4)$

Apply the distributive property.

$y - 62 = 11 x + 11 \cdot - 4$

Multiply $11$ by $- 4$ .

$y - 62 = 11 x - 44$

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

Tap for more steps...

Add $62$ to both sides of the equation.

$y = 11 x - 44 + 62$

Add $- 44$ and $62$ .

$y = 11 x + 18$

Enter YOUR Problem