Find the Difference Quotient f(x)=5x^4

$f (x) = 5 x^{4}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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

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

$f (x + h) = 5 {(x + h)}^{4}$

Simplify the result.

Use the Binomial Theorem.

$f (x + h) = 5 (x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4})$

Apply the distributive property.

$f (x + h) = 5 x^{4} + 5 (4 x^{3} h) + 5 (6 x^{2} h^{2}) + 5 (4 x h^{3}) + 5 h^{4}$

Simplify.

Multiply $4$ by $5$ .

$f (x + h) = 5 x^{4} + 20 (x^{3} h) + 5 (6 x^{2} h^{2}) + 5 (4 x h^{3}) + 5 h^{4}$

Multiply $6$ by $5$ .

$f (x + h) = 5 x^{4} + 20 (x^{3} h) + 30 (x^{2} h^{2}) + 5 (4 x h^{3}) + 5 h^{4}$

Multiply $4$ by $5$ .

$f (x + h) = 5 x^{4} + 20 (x^{3} h) + 30 (x^{2} h^{2}) + 20 (x h^{3}) + 5 h^{4}$

Remove parentheses.

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4}$

The final answer is $5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4}$ .

$5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4}$

Reorder.

Move $x^{3}$ .

$5 x^{4} + 20 h x^{3} + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4}$

Move $x^{2}$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 x h^{3} + 5 h^{4}$

Move $x$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4}$

Move $5 x^{4}$ .

$20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} + 5 x^{4}$

Move $20 h x^{3}$ .

$30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} + 20 h x^{3} + 5 x^{4}$

Move $30 h^{2} x^{2}$ .

$20 h^{3} x + 5 h^{4} + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4}$

Reorder $20 h^{3} x$ and $5 h^{4}$ .

$5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4}$

Find the components of the definition.

$f (x + h) = 5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4}$

$f (x) = 5 x^{4}$

$f (x + h) = 5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4}$

$f (x) = 5 x^{4}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - (5 x^{4})}{h}$

Step 4

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 5 x^{4}}{h}$

Subtract $5 x^{4}$ from $5 x^{4}$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 0}{h}$

Add $5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3}$ and $0$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3}}{h}$

Factor $5 h$ out of $5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3}$ .

Factor $5 h$ out of $5 h^{4}$ .

$\frac{5 h \cdot h^{3} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3}}{h}$

Factor $5 h$ out of $20 h^{3} x$ .

$\frac{5 h \cdot h^{3} + 5 h (4 h^{2} x) + 30 h^{2} x^{2} + 20 h x^{3}}{h}$

Factor $5 h$ out of $30 h^{2} x^{2}$ .

$\frac{5 h \cdot h^{3} + 5 h (4 h^{2} x) + 5 h (6 h x^{2}) + 20 h x^{3}}{h}$

Factor $5 h$ out of $20 h x^{3}$ .

$\frac{5 h \cdot h^{3} + 5 h (4 h^{2} x) + 5 h (6 h x^{2}) + 5 h (4 x^{3})}{h}$

Factor $5 h$ out of $5 h \cdot h^{3} + 5 h (4 h^{2} x)$ .

$\frac{5 h (h^{3} + 4 h^{2} x) + 5 h (6 h x^{2}) + 5 h (4 x^{3})}{h}$

Factor $5 h$ out of $5 h (h^{3} + 4 h^{2} x) + 5 h (6 h x^{2})$ .

$\frac{5 h (h^{3} + 4 h^{2} x + 6 h x^{2}) + 5 h (4 x^{3})}{h}$

Factor $5 h$ out of $5 h (h^{3} + 4 h^{2} x + 6 h x^{2}) + 5 h (4 x^{3})$ .

$\frac{5 h (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3})}{h}$

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{5 h (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3})}{h}$

Divide $5 (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3})$ by $1$ .

$5 (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3})$

Apply the distributive property.

$5 h^{3} + 5 (4 h^{2} x) + 5 (6 h x^{2}) + 5 (4 x^{3})$

Simplify.

Multiply $4$ by $5$ .

$5 h^{3} + 20 (h^{2} x) + 5 (6 h x^{2}) + 5 (4 x^{3})$

Multiply $6$ by $5$ .

$5 h^{3} + 20 (h^{2} x) + 30 (h x^{2}) + 5 (4 x^{3})$

Multiply $4$ by $5$ .

$5 h^{3} + 20 (h^{2} x) + 30 (h x^{2}) + 20 x^{3}$

Remove parentheses.

$5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3}$

Simplify the expression.

Move $h^{2}$ .

$5 h^{3} + 20 x h^{2} + 30 h x^{2} + 20 x^{3}$

Move $h$ .

$5 h^{3} + 20 x h^{2} + 30 x^{2} h + 20 x^{3}$

Move $5 h^{3}$ .

$20 x h^{2} + 30 x^{2} h + 20 x^{3} + 5 h^{3}$

Move $20 x h^{2}$ .

$30 x^{2} h + 20 x^{3} + 20 x h^{2} + 5 h^{3}$

Reorder $30 x^{2} h$ and $20 x^{3}$ .

$20 x^{3} + 30 x^{2} h + 20 x h^{2} + 5 h^{3}$

Step 5

Find the Difference Quotient f(x)=5x^4

Related Questions

Support LCMGCF.com by disabling your adblocker.