How do you use the limit definition of the derivative to find the derivative of #f(x)=x/(x+1)#?

Question

Paisley Johnson · Accepted Answer

#d/dx ( x/(x+1) ) = 1/((x+1)^2 #

By definition we have:

#f'(x) = lim_(h->0) (f(x+h)-f(x))/h#

So:

#d/dx ( x/(x+1) ) = lim_(h->0) ( (x+h)/(x+h+1) - x/(x+1)) /h#

#d/dx ( x/(x+1) ) = lim_(h->0) 1/h(( (x+h)(x+1) - x(x+h+1))/((x+1)(x+h+1))) #

#d/dx ( x/(x+1) ) = lim_(h->0) 1/h((cancelx^2+cancel(hx)+cancelx+h - cancelx^2 -cancel(hx) -cancelx)/((x+1)(x+h+1)) )#

#d/dx ( x/(x+1) ) = lim_(h->0) 1/cancel(h)(cancel(h)/((x+1)(x+h+1)) )#

#d/dx ( x/(x+1) ) = lim_(h->0) 1/((x+1)(x+h+1)) #

#d/dx ( x/(x+1) ) = 1/((x+1)^2 #

Charles Anctil · Answer

undefined To use the limit definition of the derivative to find the derivative of $ f(x) = \frac{x}{x+1} $, follow these steps:

1. Write down the limit definition of the derivative: $ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} $.
2. Substitute the function $ f(x) = \frac{x}{x+1} $ into the limit definition.
3. Expand $ f(x+h) $ by substituting $ x+h $ into the function $ \frac{x}{x+1} $.
4. Subtract $ f(x) $ from $ f(x+h) $.
5. Simplify the expression obtained from step 4.
6. Divide the result by $ h $.
7. Take the limit as $ h $ approaches 0 to find the derivative $ f'(x) $.

After following these steps, you'll find the derivative of $ f(x) = \frac{x}{x+1} $ using the limit definition.