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Problem : Compute the derivative of

g(x) =    

Applying the quotient rule yields


g'(x)=  
 =  
 =  

Problem : Find the derivative of

h(x) = ex2+sin(x)    

Let f (x) = ex, g(x) = x2 + sin(x). Then h(x) = f (g(x)), so by the chain rule,


h'(x)=f'(g(x))g'(x)  
 =(ex2+sin(x))(2x + cos(x))  

Problem : Find the derivative of g(x) = log(x), noting that g has inverse f (x) = ex and using implicit differentiation.

We differentiate both sides of f (g(x)) = x with respect to x to obtain

f'(g(x))g'(x) = 1    

or

g'(x) =    

In our case, this implies

g'(x) = =    

Figure %: Plot of g(x) = log(x) and g'(x) =
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