Question

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

Answer 1

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!