How do you find f'(x) using the definition of a derivative for #f(x)=(4/x^2) #?

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Answer 1 · 2015-10-20T20:41:40

See the explanation.

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

#f'(x)=lim_(h->0) (4/(x+h)^2-4/x^2)/h = lim_(h->0) ((4x^2-4(x+h)^2)/(x^2(x+h)^2))/h#

#f'(x)= lim_(h->0) (4x^2-4x^2-8xh-4h^2)/(hx^2(x+h)^2)#

#f'(x)= lim_(h->0) (-8xh-4h^2)/(hx^2(x+h)^2) = lim_(h->0) (-8x-4h)/(x^2(x+h)^2) = (-8x)/(x^2x^2)#

#f'(x)= -8/x^3#