If I have to use the limit definition to find the derivative and I have a f(x)= 1/square root(x+a number)....how would I start this off? Do I just leave it the same or do I have to change it? Thanks
If I have to use the limit definition to find the derivative and I have a f(x)= 1/square root(x+a number)....how would I start this off? Do I just leave it the same or do I have to change it? Thanks
Step 1:multiply by :
..........$\displaystyle \frac{\sqrt{x+a}}{\sqrt{x+a}}$ to get:
...............$\displaystyle \frac{\sqrt{x+a}}{x+a}$..............................
Step 2: form the ratio: $\displaystyle \frac{f(x+h)-f(x)}{h}$ to get,after doing some calculations:
...................$\displaystyle \frac{(x+a)\sqrt{x+h+a}-(x+h+a)\sqrt{x+a}}{h(x+a)(x+h+a)}$.................................................. ......................
Step 3: multiply by:
..............$\displaystyle \frac{(x+a)\sqrt{x+h+a}+(x+h+a)\sqrt{x+a}}{(x+a)\s qrt{x+h+a}+(x+h+a)\sqrt{x+a}}$ to get after doing some calculations:
.....................-$\displaystyle \frac{ 1}{(x+a)\sqrt{x+h+a}+(x+h+a)\sqrt{x+a}}$.................................................. ...................
Step 4: let h go to zero to get:
...................-1/2$\displaystyle \frac{ 1}{(x+a)\sqrt{x+a}}$ ,which is the desired limit............................................. .....................................