Find @FORMULA f(x)=7x^2+6

$f (x) = 7 x^{2} + 6$

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.

$f (x + h) = 7 {(x + h)}^{2} + 6$

Simplify the result.

Simplify each term.

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

$f (x + h) = 7 ((x + h) (x + h)) + 6$

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

Apply the distributive property.

$f (x + h) = 7 (x (x + h) + h (x + h)) + 6$

Apply the distributive property.

$f (x + h) = 7 (x \cdot x + x h + h (x + h)) + 6$

Apply the distributive property.

$f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = 7 (x^{2} + x h + h x + h \cdot h) + 6$

Multiply $h$ by $h$ .

$f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 6$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 7 (x^{2} + h x + h x + h^{2}) + 6$

Add $h x$ and $h x$ .

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 6$

Apply the distributive property.

$f (x + h) = 7 x^{2} + 7 (2 h x) + 7 h^{2} + 6$

Multiply $2$ by $7$ .

$f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 6$

The final answer is $7 x^{2} + 14 h x + 7 h^{2} + 6$ .

$7 x^{2} + 14 h x + 7 h^{2} + 6$

Reorder.

Move $7 x^{2}$ .

$14 h x + 7 h^{2} + 7 x^{2} + 6$

Reorder $14 h x$ and $7 h^{2}$ .

$7 h^{2} + 14 h x + 7 x^{2} + 6$

Find the components of the definition.

$f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 6$

$f (x) = 7 x^{2} + 6$

$f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 6$

$f (x) = 7 x^{2} + 6$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{7 h^{2} + 14 h x + 7 x^{2} + 6 - (7 x^{2} + 6)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 6 - (7 x^{2}) - 1 \cdot 6}{h}$

Multiply $7$ by $- 1$ .

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 6 - 7 x^{2} - 1 \cdot 6}{h}$

Multiply $- 1$ by $6$ .

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 6 - 7 x^{2} - 6}{h}$

Subtract $7 x^{2}$ from $7 x^{2}$ .

$\frac{7 h^{2} + 14 h x + 0 + 6 - 6}{h}$

Add $7 h^{2}$ and $0$ .

$\frac{7 h^{2} + 14 h x + 6 - 6}{h}$

Subtract $6$ from $6$ .

$\frac{7 h^{2} + 14 h x + 0}{h}$

Add $7 h^{2} + 14 h x$ and $0$ .

$\frac{7 h^{2} + 14 h x}{h}$

Factor $7 h$ out of $7 h^{2} + 14 h x$ .

Factor $7 h$ out of $7 h^{2}$ .

$\frac{7 h \cdot h + 14 h x}{h}$

Factor $7 h$ out of $14 h x$ .

$\frac{7 h \cdot h + 7 h (2 x)}{h}$

Factor $7 h$ out of $7 h \cdot h + 7 h (2 x)$ .

$\frac{7 h (h + 2 x)}{h}$

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{7 h (h + 2 x)}{h}$

Divide $7 (h + 2 x)$ by $1$ .

$7 (h + 2 x)$

Apply the distributive property.

$7 h + 7 (2 x)$

Simplify the expression.

Multiply $2$ by $7$ .

$7 h + 14 x$

Reorder $7 h$ and $14 x$ .

$14 x + 7 h$

Step 5

Find @FORMULA f(x)=7x^2+6

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