Question

F(x) =1/x by 1st principle of derivative

Answer 1

By first principle of differentiation, the first order derivative of a function f(x) is given by the following limit:

• ${\boxed{\bf{f^{\prime}(x)=\displaystyle \bf \lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}}}}$

So, if f(x) = (1/x), then f(x+h) = (1/x+h)

Substitute the values in the limit:

${:\implies \quad \displaystyle \sf f^{\prime}(x)=\lim_{h\to0}\dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}}$

${:\implies \quad \displaystyle \sf f^{\prime}(x)=\lim_{h\to0}\dfrac{x-x-h}{hx(x+h)}}$

${:\implies \quad \displaystyle \sf f^{\prime}(x)=\lim_{h\to0}\dfrac{-1}{x(x+h)}}$

${:\implies \quad \boxed{\bf{f^{\prime}(x)=-\dfrac{1}{x^2}}}}$