Calculus Examples

$f (x) = 2 x^{3}$

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.

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

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Use the Binomial Theorem.

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

Step 2.1.2.2

Apply the distributive property.

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

Step 2.1.2.3

Simplify.

Tap for more steps...

Step 2.1.2.3.1

Multiply $3$ by $2$ .

$f (x + h) = 2 x^{3} + 6 (x^{2} h) + 2 (3 x h^{2}) + 2 h^{3}$

Step 2.1.2.3.2

Multiply $3$ by $2$ .

$f (x + h) = 2 x^{3} + 6 (x^{2} h) + 6 (x h^{2}) + 2 h^{3}$

Step 2.1.2.4

Remove parentheses.

$f (x + h) = 2 x^{3} + 6 x^{2} h + 6 x h^{2} + 2 h^{3}$

Step 2.1.2.5

The final answer is $2 x^{3} + 6 x^{2} h + 6 x h^{2} + 2 h^{3}$ .

$2 x^{3} + 6 x^{2} h + 6 x h^{2} + 2 h^{3}$

Step 2.2

Reorder.

Tap for more steps...

Step 2.2.1

Move $x^{2}$ .

$2 x^{3} + 6 h x^{2} + 6 x h^{2} + 2 h^{3}$

Step 2.2.2

Move $x$ .

$2 x^{3} + 6 h x^{2} + 6 h^{2} x + 2 h^{3}$

Step 2.2.3

Move $2 x^{3}$ .

$6 h x^{2} + 6 h^{2} x + 2 h^{3} + 2 x^{3}$

Step 2.2.4

Move $6 h x^{2}$ .

$6 h^{2} x + 2 h^{3} + 6 h x^{2} + 2 x^{3}$

Step 2.2.5

Reorder $6 h^{2} x$ and $2 h^{3}$ .

$2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3}$

Step 2.3

Find the components of the definition.

$f (x + h) = 2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3}$

$f (x) = 2 x^{3}$

$f (x + h) = 2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3}$

$f (x) = 2 x^{3}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3} - (2 x^{3})}{h}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Multiply $- 1$ by $2$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3} - 2 x^{3}}{h}$

Step 4.1.2

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{3} + 6 h^{2} x + 6 h x^{2} + 0}{h}$

Step 4.1.3

Add $2 h^{3} + 6 h^{2} x + 6 h x^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{3} + 6 h^{2} x + 6 h x^{2}}{h}$

Step 4.1.4

Factor $2 h$ out of $2 h^{3} + 6 h^{2} x + 6 h x^{2}$ .

Tap for more steps...

Step 4.1.4.1

Factor $2 h$ out of $2 h^{3}$ .

$f' (x) = lim_{h \to 0} \frac{2 h \cdot h^{2} + 6 h^{2} x + 6 h x^{2}}{h}$

Step 4.1.4.2

Factor $2 h$ out of $6 h^{2} x$ .

$f' (x) = lim_{h \to 0} \frac{2 h \cdot h^{2} + 2 h (3 h x) + 6 h x^{2}}{h}$

Step 4.1.4.3

Factor $2 h$ out of $6 h x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{2 h \cdot h^{2} + 2 h (3 h x) + 2 h (3 x^{2})}{h}$

Step 4.1.4.4

Factor $2 h$ out of $2 h \cdot h^{2} + 2 h (3 h x)$ .

$f' (x) = lim_{h \to 0} \frac{2 h (h^{2} + 3 h x) + 2 h (3 x^{2})}{h}$

Step 4.1.4.5

Factor $2 h$ out of $2 h (h^{2} + 3 h x) + 2 h (3 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{2 h (h^{2} + 3 h x + 3 x^{2})}{h}$

Step 4.2

Simplify terms.

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.

$f' (x) = lim_{h \to 0} \frac{2 h (h^{2} + 3 h x + 3 x^{2})}{h}$

Step 4.2.1.2

Divide $2 (h^{2} + 3 h x + 3 x^{2})$ by $1$ .

$f' (x) = lim_{h \to 0} 2 (h^{2} + 3 h x + 3 x^{2})$

Step 4.2.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} 2 h^{2} + 2 (3 h x) + 2 (3 x^{2})$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Multiply $3$ by $2$ .

$f' (x) = lim_{h \to 0} 2 h^{2} + 6 (h x) + 2 (3 x^{2})$

Step 4.3.2

Multiply $3$ by $2$ .

$f' (x) = lim_{h \to 0} 2 h^{2} + 6 (h x) + 6 x^{2}$

Step 4.4

Move $h$ .

$f' (x) = lim_{h \to 0} 2 h^{2} + 6 x h + 6 x^{2}$

Step 4.5

Move $2 h^{2}$ .

$f' (x) = lim_{h \to 0} 6 x h + 6 x^{2} + 2 h^{2}$

Step 4.6

Reorder $6 x h$ and $6 x^{2}$ .

$f' (x) = lim_{h \to 0} 6 x^{2} + 6 x h + 2 h^{2}$

Step 5

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

$lim_{h \to 0} 6 x^{2} + lim_{h \to 0} 6 x h + lim_{h \to 0} 2 h^{2}$

Step 6

Evaluate the limit of $6 x^{2}$ which is constant as $h$ approaches $0$ .

$6 x^{2} + lim_{h \to 0} 6 x h + lim_{h \to 0} 2 h^{2}$

Step 7

Move the term $6 x$ outside of the limit because it is constant with respect to $h$ .

$6 x^{2} + 6 x lim_{h \to 0} h + lim_{h \to 0} 2 h^{2}$

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

$6 x^{2} + 6 x lim_{h \to 0} h + 2 lim_{h \to 0} h^{2}$

Step 9

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$6 x^{2} + 6 x lim_{h \to 0} h + 2 {(lim_{h \to 0} h)}^{2}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

Tap for more steps...

Step 10.1

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

$6 x^{2} + 6 x \cdot 0 + 2 {(lim_{h \to 0} h)}^{2}$

Step 10.2

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

$6 x^{2} + 6 x \cdot 0 + 2 \cdot 0^{2}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Multiply $6 x \cdot 0$ .

Tap for more steps...

Step 11.1.1.1

Multiply $0$ by $6$ .

$6 x^{2} + 0 x + 2 \cdot 0^{2}$

Step 11.1.1.2

Multiply $0$ by $x$ .

$6 x^{2} + 0 + 2 \cdot 0^{2}$

Step 11.1.2

Raising $0$ to any positive power yields $0$ .

$6 x^{2} + 0 + 2 \cdot 0$

Step 11.1.3

Multiply $2$ by $0$ .

$6 x^{2} + 0 + 0$

Step 11.2

Combine the opposite terms in $6 x^{2} + 0 + 0$ .

Tap for more steps...

Step 11.2.1

Add $6 x^{2}$ and $0$ .

$6 x^{2} + 0$

Step 11.2.2

Add $6 x^{2}$ and $0$ .

$6 x^{2}$

Step 12