Question

for the function ( f(x)=4 x^{2}-5 x ), evaluate and simplify.
( \frac{f(x+h)-f(x)}{h}= )
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Explanation:

Step1: Find \(f(x + h)\)

Substitute \(x+h\) into \(f(x)\):
\(f(x + h)=4(x + h)^{2}-5(x + h)\)
Expand \((x + h)^{2}\):
\(f(x + h)=4(x^{2}+2xh+h^{2})-5x - 5h\)
\(f(x + h)=4x^{2}+8xh+4h^{2}-5x - 5h\)

Step2: Calculate \(f(x + h)-f(x)\)

\(f(x + h)-f(x)=(4x^{2}+8xh+4h^{2}-5x - 5h)-(4x^{2}-5x)\)
Remove the parentheses:
\(f(x + h)-f(x)=4x^{2}+8xh+4h^{2}-5x - 5h - 4x^{2}+5x\)
Simplify:
\(f(x + h)-f(x)=8xh+4h^{2}-5h\)

Step3: Calculate \(\frac{f(x + h)-f(x)}{h}\)

\(\frac{f(x + h)-f(x)}{h}=\frac{8xh+4h^{2}-5h}{h}\)
Factor out \(h\) in the numerator:
\(\frac{f(x + h)-f(x)}{h}=\frac{h(8x + 4h-5)}{h}\)
Cancel out \(h\) (\(h
eq0\)):
\(\frac{f(x + h)-f(x)}{h}=8x + 4h-5\)

Answer:

\(8x + 4h-5\)