Simplifying calculus by simplifying equations at each step

By Tutor GuyNo Comments


Get in the habit of simplifying your equations as you go. If you need to find a second (or higher) derivative, simplify f ′(x) before you take the derivative again. If you are finding the volume of revolution, simplify your integral before you evaluate it. If you are applying l’Hôpital’s rule, simplify your new expression before you plug in the value of x again. If you don’t simplify, you usually make your task much harder.

Example: Find f ″(x).

f(x)= \dfrac{2x+1}{x+1}

 Solution: The first derivative is straightforward:

f'(x)= \dfrac{(x+1)2-(2x+1)1}{(x+1)^2}

Taking the derivative of this is somewhat messy, unless you simplify first, as follows:

f'(x)= \dfrac{(x+1)2-(2x+1)1}{(x+1)^2}= \dfrac{(2x+2)-(2x+1)}{(x+1)^2}= \dfrac{1}{(x+1)^2}

Now the second derivative is much easier to evaluate:

f''(x)= \dfrac{-2}{(x+1)^3}

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