Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 8


If \displaystyle f\left(x\right)=x^2+1, find \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0.


Solution:

\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{\left[\left(x+h\right)^2+1\right]-\left(x^2+1\right)\:}{h}

\displaystyle =\frac{x^2+2xh+h^2+1-x^2-1}{h}

\displaystyle =\frac{2xh+h^2}{h}

\displaystyle =\frac{h\left(2x+h\right)}{h}

\displaystyle =2x+h


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