Calculus Examples

$y = x^{2} - x$ , $(1, 0)$

Find $\frac{d y}{d x}$ and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line at $x = 1$ and $y = 0$ .

Tap for more steps...

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

Tap for more steps...

Differentiate.

Tap for more steps...

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

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

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

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

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

Tap for more steps...

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

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

$2 x - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$2 x - 1$

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

$y' = 2 x - 1$

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

$\frac{d y}{d x} = 2 x - 1$

Evaluate at $x = 1$ and $y = 0$ .

Tap for more steps...

Replace the variable $x$ with $1$ in the expression.

$2 (1) - 1$

Multiply $2$ by $1$ .

$2 - 1$

Subtract $1$ from $2$ .

$1$

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

$y - (0) = 1 \cdot (x - (1))$

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...

Subtract $0$ from $y$ .

$y = 1 \cdot (x - (1))$

Simplify $1 \cdot (x - (1))$ .

Tap for more steps...

Multiply $x - (1)$ by $1$ .

$y = x - (1)$

Multiply $- 1$ by $1$ .

$y = x - 1$

Enter YOUR Problem