Find dy/dx using first principle x+1/x

Question

asked Jan 5, 2022 126k views

1 Answer

Gregory Saxton · Answer 1 · 2022-01-10T13:03:51+0000

Answer:

The derivative of the function is:

f'(x)=1-(1)/(x^(2))

Explanation:

The first principle is given by:

$f'(x)=(dy)/(dx)=lim_(h\rightarrow 0)(f(x+h)-f(x))/(h)$

The function here is:

f(x)=x+(1)/(x)

Now, using the first principle we have:

$f'(x)=lim_(h\rightarrow 0)((x+h)+(1)/((x+h))-x-(1)/(x))/(h)$

$=lim_(h\rightarrow 0)(h+(1)/((x+h))-(1)/(x))/(h)$

$=lim_(h\rightarrow 0)(h-(h)/(x(x+h)))/(h)$

$=lim_(h\rightarrow 0)(h(1-(1)/(x(x+h))))/(h)$

$=lim_(h\rightarrow 0)1-(1)/(x(x+h))$

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

Therefore, the derivative of the function is:

f'(x)=1-(1)/(x^(2))

I hope it helps you!