Question

find the difference quotient of ( f(x) = x^2 - 5 ); that is find ( \frac{f(x+h) - f(x)}{h} ), ( h
eq 0 ). be sure to simplify. the difference quotient is

Explanation:

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

Given \( f(x)=x^{2}-5 \), substitute \( x + h \) into \( f(x) \):
\( f(x + h)=(x + h)^{2}-5 \)
Expand \( (x + h)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a = x \) and \( b = h \):
\( f(x + h)=x^{2}+2xh+h^{2}-5 \)

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

Substitute \( f(x + h)=x^{2}+2xh + h^{2}-5 \) and \( f(x)=x^{2}-5 \) into \( f(x + h)-f(x) \):
\( f(x + h)-f(x)=(x^{2}+2xh + h^{2}-5)-(x^{2}-5) \)
Simplify by removing the parentheses and combining like terms:
\( f(x + h)-f(x)=x^{2}+2xh + h^{2}-5 - x^{2}+5 \)
The \( x^{2} \) and \( -x^{2} \), \( -5 \) and \( +5 \) cancel out, leaving:
\( f(x + h)-f(x)=2xh+h^{2} \)

Step3: Compute the difference quotient \( \frac{f(x + h)-f(x)}{h} \)

Substitute \( f(x + h)-f(x)=2xh + h^{2} \) into the difference quotient:
\( \frac{f(x + h)-f(x)}{h}=\frac{2xh + h^{2}}{h} \)
Factor out \( h \) from the numerator:
\( \frac{f(x + h)-f(x)}{h}=\frac{h(2x + h)}{h} \)
Since \( h
eq0 \), we can cancel out the \( h \) in the numerator and the denominator:
\( \frac{f(x + h)-f(x)}{h}=2x + h \)

Answer:

\( 2x + h \)