Question

given the function $f(x) = -5x^2 + 4$, express the value of $\frac{f(x+h)-f(x)}{h}$ in simplest form.

Explanation:

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

Substitute \( x + h \) into \( f(x) \):
\( f(x + h)=-5(x + h)^{2}+4 \)
Expand \( (x + h)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), here \( a = x \), \( b = h \), so \( (x + h)^{2}=x^{2}+2xh+h^{2} \)
Then \( f(x + h)=-5(x^{2}+2xh + h^{2})+4=-5x^{2}-10xh-5h^{2}+4 \)

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

Substitute \( f(x + h) \) and \( f(x) \) into the expression:
\( f(x + h)-f(x)=(-5x^{2}-10xh - 5h^{2}+4)-(-5x^{2}+4) \)
Remove the parentheses: \( -5x^{2}-10xh-5h^{2}+4 + 5x^{2}-4 \)
Combine like terms: \( (-5x^{2}+5x^{2})-10xh-5h^{2}+(4 - 4)=-10xh-5h^{2} \)

Step3: Divide by \( h \) ( \( h

eq0 \))
\( \frac{f(x + h)-f(x)}{h}=\frac{-10xh-5h^{2}}{h} \)
Factor out \( h \) from the numerator: \( \frac{h(-10x - 5h)}{h} \)
Cancel out the common factor \( h \) (since \( h
eq0 \)): \( -10x-5h \)

Answer:

\( -10x - 5h \)