derivatives problem using Definition of Derivatives (formula)

Question

Given f(x) 1/(2x²+5) what will be f^'(x) using the definition of derivatives (formula)?

Answer 1

By the definition the derivative of f(x) is 

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

In your example, f(x)=1/(2x²+5). So f(x+h)=1/(2(x+h)²+5)

and f(x+h)-f(x)=1/(2(x+h)²+5)-1/(2x²+5)=-2h(2x+h)/[2(x+h)²+5)(2x²+5)].

Now by the above formula (in the dfinition)

f'(x)=-2lim_x-->0(2x+h)/[(2(x+h)²+5)(2x²+5)]=-4x/(2x²+5)².