Answer :
We are given the following function:
[tex]f\mleft(x\mright)=x^2+9[/tex]We are asked to determine the following quotient:
[tex]\frac{f(x+h)-f(x)}{h}[/tex]To do that, we will first determine the function f(x + h), to do that we will replace the variable "x" for the variable "x + h", like this:
[tex]f(x+h)=(x+h)^2+9[/tex]Solving the parenthesis, using the following property:
[tex](a+b)^2=a^2+2ab+b^2[/tex]Applying the property:
[tex]f(x+h)=x^2+2hx+h^2+9[/tex]Substituting in the quotient:
[tex]\frac{x^2+2hx+h^2+9-(x^2+9)}{h}[/tex]Now we change the sing to the terms inside the parenthesis since it is preceded by a minus sing:
[tex]\frac{x^2+2hx+h^2+9-x^2-9}{h}[/tex]Adding line terms:
[tex]\frac{2hx+h^2}{h}[/tex]Now we take "h" as a common factor in the numerator:
[tex]\frac{h(2x+h)}{h}[/tex]Canceling out the "h":
[tex]2x+h[/tex]