Calculus Examples

$y = 4 x - 3 x^{2}$

Step 1

Reorder $4 x$ and $- 3 x^{2}$ .

$y = - 3 x^{2} + 4 x$

Step 2

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

$f (x) = - 3 x^{2} + 4 x$

Step 3

Find the derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $- 3 x^{2} + 4 x$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [4 x]$ .

$\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [4 x]$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x]$

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $- 3$ .

$- 6 x + \frac{d}{d x} [4 x]$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$- 6 x + 4 \frac{d}{d x} [x]$

Step 3.3.2

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

$- 6 x + 4 \cdot 1$

Step 3.3.3

Multiply $4$ by $1$ .

$- 6 x + 4$

Step 4

Set the derivative equal to $0$ then solve the equation $- 6 x + 4 = 0$ .

Tap for more steps...

Step 4.1

Subtract $4$ from both sides of the equation.

$- 6 x = - 4$

Step 4.2

Divide each term in $- 6 x = - 4$ by $- 6$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $- 6 x = - 4$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 4}{- 6}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $- 6$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 4}{- 6}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 6}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Cancel the common factor of $- 4$ and $- 6$ .

Tap for more steps...

Step 4.2.3.1.1

Factor $- 2$ out of $- 4$ .

$x = \frac{- 2 (2)}{- 6}$

Step 4.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.1.2.1

Factor $- 2$ out of $- 6$ .

$x = \frac{- 2 \cdot 2}{- 2 \cdot 3}$

Step 4.2.3.1.2.2

Cancel the common factor.

$x = \frac{- 2 \cdot 2}{- 2 \cdot 3}$

Step 4.2.3.1.2.3

Rewrite the expression.

$x = \frac{2}{3}$

Step 5

Solve the original function $f (x) = - 3 x^{2} + 4 x$ at $x = \frac{2}{3}$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with $\frac{2}{3}$ in the expression.

$f (\frac{2}{3}) = - 3 {(\frac{2}{3})}^{2} + 4 (\frac{2}{3})$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Apply the product rule to $\frac{2}{3}$ .

$f (\frac{2}{3}) = - 3 \frac{2^{2}}{3^{2}} + 4 (\frac{2}{3})$

Step 5.2.1.2

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

$f (\frac{2}{3}) = - 3 \frac{4}{3^{2}} + 4 (\frac{2}{3})$

Step 5.2.1.3

Raise $3$ to the power of $2$ .

$f (\frac{2}{3}) = - 3 (\frac{4}{9}) + 4 (\frac{2}{3})$

Step 5.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.1.4.1

Factor $3$ out of $- 3$ .

$f (\frac{2}{3}) = 3 (- 1) (\frac{4}{9}) + 4 (\frac{2}{3})$

Step 5.2.1.4.2

Factor $3$ out of $9$ .

$f (\frac{2}{3}) = 3 \cdot (- 1 \frac{4}{3 \cdot 3}) + 4 (\frac{2}{3})$

Step 5.2.1.4.3

Cancel the common factor.

$f (\frac{2}{3}) = 3 \cdot (- 1 \frac{4}{3 \cdot 3}) + 4 (\frac{2}{3})$

Step 5.2.1.4.4

Rewrite the expression.

$f (\frac{2}{3}) = - 1 (\frac{4}{3}) + 4 (\frac{2}{3})$

Step 5.2.1.5

Rewrite $- 1 (\frac{4}{3})$ as $- (\frac{4}{3})$ .

$f (\frac{2}{3}) = - (\frac{4}{3}) + 4 (\frac{2}{3})$

Step 5.2.1.6

Multiply $4 (\frac{2}{3})$ .

Tap for more steps...

Step 5.2.1.6.1

Combine $4$ and $\frac{2}{3}$ .

$f (\frac{2}{3}) = - \frac{4}{3} + \frac{4 \cdot 2}{3}$

Step 5.2.1.6.2

Multiply $4$ by $2$ .

$f (\frac{2}{3}) = - \frac{4}{3} + \frac{8}{3}$

Step 5.2.2

Combine fractions.

Tap for more steps...

Step 5.2.2.1

Combine the numerators over the common denominator.

$f (\frac{2}{3}) = \frac{- 4 + 8}{3}$

Step 5.2.2.2

Add $- 4$ and $8$ .

$f (\frac{2}{3}) = \frac{4}{3}$

Step 5.2.3

The final answer is $\frac{4}{3}$ .

$\frac{4}{3}$

Step 6

The horizontal tangent line on function $f (x) = - 3 x^{2} + 4 x$ is $y = \frac{4}{3}$ .

$y = \frac{4}{3}$

Step 7