Find the Difference Quotient f(x)=3x+6

Solution:

Step 1:

Utilize the standard formula for the difference quotient: $\frac{f(x+h)-f(x)}{h}$.

Step 2:

Determine the individual elements of the formula.

Step 2.1:

Calculate $f(x+h)$.

Step 2.1.1:

Substitute $x$ with $x+h$ in $f(x)$. Thus, $f(x+h)=3(x+h)+6$.

Step 2.1.2:

Streamline the expression.

Step 2.1.2.1:

Use the distributive law: $f(x+h)=3x+3h+6$.

Step 2.1.2.2:

The simplified form is $3x+3h+6$.

Step 2.2:

Rearrange the terms: $3h+3x+6$.

Step 2.3:

Identify the components for $f(x+h)$ and $f(x)$: $f(x+h)=3h+3x+6$ and $f(x)=3x+6$.

Step 3:

Insert the identified components into the difference quotient: $\frac{f(x+h)-f(x)}{h}=\frac{3h+3x+6-(3x+6)}{h}$.

Step 4:

Simplify the expression.

Step 4.1:

Condense the numerator.

Step 4.1.1:

Extract the common factor of $3$ from $3x+6$.

Step 4.1.1.1:

Extract $3$ from $3x$: $\frac{3h+3x+6-(3(x)+6)}{h}$.

Step 4.1.1.2:

Extract $3$ from $6$: $\frac{3h+3x+6-(3x+3\cdot 2)}{h}$.

Step 4.1.1.3:

Extract $3$ from $3x+3\cdot 2$: $\frac{3h+3x+6-3(x+2)}{h}$.

Step 4.1.2:

Distribute the negative sign across $3$: $\frac{3h+3x+6-3(x+2)}{h}$.

Step 4.1.3:

Extract the factor of $3$ from the entire expression.

Step 4.1.3.1:

Extract $3$ from $3h$: $\frac{3h+3x+6-3(x+2)}{h}$.

Step 4.1.3.2:

Extract $3$ from $3x$: $\frac{3h+3(x)+6-3(x+2)}{h}$.

Step 4.1.3.3:

Extract $3$ from $6$: $\frac{3h+3(x)+3(2)-3(x+2)}{h}$.

Step 4.1.3.4:

Extract $3$ from $-3(x+2)$: $\frac{3h+3(x)+3(2)-3(-(x+2))}{h}$.

Step 4.1.3.5:

Extract $3$ from $3h+3(x)$: $\frac{3(h+x)+3(2)-3(-(x+2))}{h}$.

Step 4.1.3.6:

Extract $3$ from $3(h+x)+3(2)$: $\frac{3(h+x+2)-3(-(x+2))}{h}$.

Step 4.1.3.7:

Extract $3$ from $3(h+x+2)-3(-(x+2))$: $\frac{3(h+x+2-(x+2))}{h}$.

Step 4.1.4:

Apply the distributive property: $\frac{3(h+x+2-x-2)}{h}$.

Step 4.1.5:

Distribute the negative sign across $2$: $\frac{3(h+x+2-x-2)}{h}$.

Step 4.1.6:

Eliminate $x$ from $x$: $\frac{3(h+0+2-2)}{h}$.

Step 4.1.7:

Combine $h$ and $0$: $\frac{3(h+2-2)}{h}$.

Step 4.1.8:

Subtract $2$ from $2$: $\frac{3(h+0)}{h}$.

Step 4.1.9:

Combine $h$ and $0$: $\frac{3h}{h}$.

Step 4.2:

Eliminate the common $h$ factor.

Step 4.2.1:

Cancel out $h$: $\frac{3\cancel{h}}{\cancel{h}}$.

Step 4.2.2:

Simplify $3$ over $1$: $3$.

Step 5:

The final result is $3$.

Knowledge Notes:

The difference quotient is a fundamental concept in calculus, used to define the derivative of a function. The formula for the difference quotient is given by $\frac{f(x+h)-f(x)}{h}$, where $h$ is a small increment in $x$, and $f(x)$ is a function of $x$.

In this problem, we applied the difference quotient to the linear function $f(x)=3x+6$. The process involved substituting $x+h$ into the function, simplifying the resulting expression, and then plugging it into the difference quotient formula. The simplification steps required the use of the distributive property and combining like terms.

The final step was to cancel the common factors in the numerator and denominator, which is a common technique in algebra to simplify fractions. The result of the difference quotient for a linear function is the slope of the function, which in this case is $3$. This is consistent with the fact that the derivative of a linear function is equal to its slope.