Calculus Examples

$y = 4 x - x^{2}$

Step 1

Reorder $4 x$ and $- x^{2}$ .

$y = - x^{2} + 4 x$

Step 2

Set $y$ as a function of $x$ .

$f (x) = - x^{2} + 4 x$

Step 3

Find the derivative.

Tap for more steps...

By the Sum Rule, the derivative of $- x^{2} + 4 x$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4 x]$ .

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

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

Tap for more steps...

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

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

$- (2 x) + \frac{d}{d x} [4 x]$

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [4 x]$

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

Tap for more steps...

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

$- 2 x + 4 \frac{d}{d x} [x]$

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

$- 2 x + 4 \cdot 1$

Multiply $4$ by $1$ .

$- 2 x + 4$

Step 4

Set the derivative equal to $0$ then solve the equation $- 2 x + 4 = 0$ .

Tap for more steps...

Subtract $4$ from both sides of the equation.

$- 2 x = - 4$

Divide each term in $- 2 x = - 4$ by $- 2$ and simplify.

Tap for more steps...

Divide each term in $- 2 x = - 4$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 4}{- 2}$

Simplify the left side.

Tap for more steps...

Cancel the common factor of $- 2$ .

Tap for more steps...

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 4}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 2}$

Simplify the right side.

Tap for more steps...

Divide $- 4$ by $- 2$ .

$x = 2$

Step 5

Solve the original function $f (x) = - x^{2} + 4 x$ at $x = 2$ .

Tap for more steps...

Replace the variable $x$ with $2$ in the expression.

$f (2) = - {(2)}^{2} + 4 (2)$

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

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

$f (2) = - 1 \cdot 4 + 4 (2)$

Multiply $- 1$ by $4$ .

$f (2) = - 4 + 4 (2)$

Multiply $4$ by $2$ .

$f (2) = - 4 + 8$

Add $- 4$ and $8$ .

$f (2) = 4$

The final answer is $4$ .

$4$

Step 6

The horizontal tangent line on function $f (x) = - x^{2} + 4 x$ is $y = 4$ .

$y = 4$

Step 7