Calculus Examples

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

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

Evaluate the function at $x = x + h$ .

Tap for more steps...

Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

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

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

Simplify each term.

Tap for more steps...

Step 2.1.2.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = - ((x + h) (x + h)) + x + h$

Step 2.1.2.2.2

Expand $(x + h) (x + h)$ using the FOIL Method.

Tap for more steps...

Step 2.1.2.2.2.1

Apply the distributive property.

$f (x + h) = - (x (x + h) + h (x + h)) + x + h$

Step 2.1.2.2.2.2

Apply the distributive property.

$f (x + h) = - (x \cdot x + x h + h (x + h)) + x + h$

Step 2.1.2.2.2.3

Apply the distributive property.

$f (x + h) = - (x \cdot x + x h + h x + h \cdot h) + x + h$

Step 2.1.2.2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.2.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = - (x^{2} + x h + h x + h \cdot h) + x + h$

Step 2.1.2.2.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.2.3.2

Add $x h$ and $h x$ .

Tap for more steps...

Step 2.1.2.2.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.2.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.2.4

Apply the distributive property.

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

Step 2.1.2.2.5

Multiply $2$ by $- 1$ .

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

Step 2.1.2.3

The final answer is $- x^{2} - 2 h x - h^{2} + x + h$ .

$- x^{2} - 2 h x - h^{2} + x + h$

Step 2.2

Reorder.

Tap for more steps...

Step 2.2.1

Move $x$ .

$- x^{2} - 2 h x - h^{2} + h + x$

Step 2.2.2

Move $- x^{2}$ .

$- 2 h x - h^{2} - x^{2} + h + x$

Step 2.2.3

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

$- h^{2} - 2 h x - x^{2} + h + x$

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + h + x - (- x^{2} + x)}{h}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + h + x + x^{2} - x}{h}$

Step 4.1.2

Multiply $- - x^{2}$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + h + x + 1 x^{2} - x}{h}$

Step 4.1.2.2

Multiply $x^{2}$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + h + x + x^{2} - x}{h}$

Step 4.1.3

Add $- x^{2}$ and $x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x + h + x + 0 - x}{h}$

Step 4.1.4

Add $- h^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x + h + x - x}{h}$

Step 4.1.5

Subtract $x$ from $x$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x + h + 0}{h}$

Step 4.1.6

Add $- h^{2} - 2 h x + h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x + h}{h}$

Step 4.1.7

Factor $h$ out of $- h^{2} - 2 h x + h$ .

Tap for more steps...

Step 4.1.7.1

Factor $h$ out of $- h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (- h) - 2 h x + h}{h}$

Step 4.1.7.2

Factor $h$ out of $- 2 h x$ .

$f' (x) = lim_{h \to 0} \frac{h (- h) + h (- 2 x) + h}{h}$

Step 4.1.7.3

Raise $h$ to the power of $1$ .

$f' (x) = lim_{h \to 0} \frac{h (- h) + h (- 2 x) + h}{h}$

Step 4.1.7.4

Factor $h$ out of $h^{1}$ .

$f' (x) = lim_{h \to 0} \frac{h (- h) + h (- 2 x) + h \cdot 1}{h}$

Step 4.1.7.5

Factor $h$ out of $h (- h) + h (- 2 x)$ .

$f' (x) = lim_{h \to 0} \frac{h (- h - 2 x) + h \cdot 1}{h}$

Step 4.1.7.6

Factor $h$ out of $h (- h - 2 x) + h \cdot 1$ .

$f' (x) = lim_{h \to 0} \frac{h (- h - 2 x + 1)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (- h - 2 x + 1)}{h}$

Step 4.2.1.2

Divide $- h - 2 x + 1$ by $1$ .

$f' (x) = lim_{h \to 0} - h - 2 x + 1$

Step 4.2.2

Reorder $- h$ and $- 2 x$ .

$f' (x) = lim_{h \to 0} - 2 x - h + 1$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$- lim_{h \to 0} 2 x - lim_{h \to 0} h + lim_{h \to 0} 1$

Step 6

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

$- 2 x - lim_{h \to 0} h + lim_{h \to 0} 1$

Step 7

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$- 2 x - lim_{h \to 0} h + 1$

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$- 2 x + 0 + 1$

Step 8.2

Add $- 2 x$ and $0$ .

$- 2 x + 1$

Step 9