Find derivitive using limit process
asked Sep 24, 2016 in Calculus Answers by anonymous

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f(x)=4/x^(1/2); f(x+h)=4/(x+h)^(1/2); differential is f'(x)=(f(x+h)-f(x))/h.

4/(x+h)^(1/2)=4/(x^(1/2)(1+h/x)^(1/2))=(4x^(-1/2)) * (1+h/x)^(-1/2).

(1+h/x)^(-1/2) can be expanded using the binomial theorem:

1-h/2x+(-1/2)(-3/2)h^2/2x^2+...

But if h is very small we can ignore higher powers so that this comes to 1-h/2x.

4/(x+h)^(1/2)=(4x^(-1/2)) * (1-h/2x).

So we have f'(x)=(f(x+h)-f(x))/h=((4x^(-1/2)) * (1-h/2x)-4x^(-1/2))/h=

(4x^(-1/2)-4x^(-1/2)(h/2x)-4x^(-1/2))/h=-4x^(-1/2)(1/2x)=

-4x^(-1/2-1)/2=-2x^(-3/2).

Therefore the derivative as h approaches zero is f'(x)=-2x^(-3/2).

 

answered Sep 24, 2016 by Rod Top Rated User (424,900 points)
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