Find @FORMULA f(x)=7-x^2

$f (x) = 7 - x^{2}$

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.

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

Simplify the result.

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

The final answer is $7 - x^{2} - 2 h x - h^{2}$ .

$7 - x^{2} - 2 h x - h^{2}$

Reorder.

Move $7$ .

$- x^{2} - 2 h x - h^{2} + 7$

Move $- x^{2}$ .

$- 2 h x - h^{2} - x^{2} + 7$

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

$- h^{2} - 2 h x - x^{2} + 7$

Find the components of the definition.

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

$f (x) = 7 - x^{2}$

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

$f (x) = 7 - x^{2}$

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $7$ .

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

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

Subtract $7$ from $7$ .

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

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

$\frac{- h^{2} - 2 h x}{h}$

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

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

$\frac{h (- h) - 2 h x}{h}$

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

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

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

$\frac{h (- h - 2 x)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (- h - 2 x)}{h}$

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

$- h - 2 x$

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

$- 2 x - h$

Step 5

Find @FORMULA f(x)=7-x^2

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