Question

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

Explanation:

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

Substitute \( x + h \) into \( f(x) = 5x^2 + x \).
\( f(x + h)=5(x + h)^2+(x + h) \)
Expand \( (x + h)^2 \): \( (x + h)^2=x^2 + 2xh+h^2 \)
So \( f(x + h)=5(x^2 + 2xh + h^2)+x + h=5x^2+10xh + 5h^2+x + h \)

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

Subtract \( f(x)=5x^2 + x \) from \( f(x + h) \).
\( f(x + h)-f(x)=(5x^2+10xh + 5h^2+x + h)-(5x^2 + x) \)
Simplify: \( 5x^2+10xh + 5h^2+x + h - 5x^2 - x=10xh+5h^2 + h \)

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

eq0 \))
\( \frac{f(x + h)-f(x)}{h}=\frac{10xh + 5h^2+h}{h} \)
Factor out \( h \) from numerator: \( \frac{h(10x + 5h + 1)}{h} \)
Cancel \( h \): \( 10x + 5h+1 \)

Answer:

\( 10x + 5h + 1 \)