Of course! Let’s find the expression for [tex]\(\frac{f(x+\Delta x) - f(x)}{\Delta x}\)[/tex] given that [tex]\( f(x) = x^3 \)[/tex].
### Step-by-Step Solution:
1. Function Definition:
We have [tex]\( f(x) = x^3 \)[/tex].
2. Increment the Input:
Increment the function's input by [tex]\(\Delta x\)[/tex], giving us [tex]\( f(x + \Delta x) \)[/tex].
3. Apply the Increment:
So, [tex]\( f(x + \Delta x) = (x + \Delta x)^3 \)[/tex].
4. Expand the Expression:
Next, we need to expand [tex]\( (x + \Delta x)^3 \)[/tex].
[tex]\[
(x + \Delta x)^3 = x^3 + 3x^2 (\Delta x) + 3x (\Delta x)^2 + (\Delta x)^3
\][/tex]
5. Substitute into the Difference Quotient:
We now substitute [tex]\( f(x + \Delta x) \)[/tex] and [tex]\( f(x) \)[/tex] into the difference quotient [tex]\(\frac{f(x+\Delta x) - f(x)}{\Delta x} \)[/tex].
[tex]\[
\frac{(x + \Delta x)^3 - x^3}{\Delta x}
\][/tex]
6. Calculate the Numerator:
Substitute the expanded form:
[tex]\[
\frac{(x^3 + 3x^2 (\Delta x) + 3x (\Delta x)^2 + (\Delta x)^3) - x^3}{\Delta x}
\][/tex]
Simplifying the numerator:
[tex]\[
\frac{x^3 + 3x^2 (\Delta x) + 3x (\Delta x)^2 + (\Delta x)^3 - x^3}{\Delta x}
= \frac{3x^2 (\Delta x) + 3x (\Delta x)^2 + (\Delta x)^3}{\Delta x}
\][/tex]
7. Factor and Simplify:
Factor out [tex]\(\Delta x\)[/tex] from the numerator:
[tex]\[
\frac{\Delta x (3x^2 + 3x (\Delta x) + (\Delta x)^3)}{\Delta x}
\][/tex]
Since [tex]\(\Delta x \neq 0\)[/tex], we can cancel [tex]\(\Delta x\)[/tex] in the numerator and denominator:
[tex]\[
3x^2 + 3x (\Delta x) + (\Delta x)^2
\][/tex]
8. Result:
Therefore, the expression [tex]\(\frac{f(x+\Delta x)-f(x)}{\Delta x}\)[/tex] simplifies to:
[tex]\[
\boxed{3x^2 + 3x (\Delta x) + (\Delta x)^2}
\][/tex]
So, the expression after simplifying is [tex]\( \frac{f(x + \Delta x) - f(x)}{\Delta x} = 3x^2 + 3x (\Delta x) + (\Delta x)^2 \)[/tex].
