Math-Linux.com

Knowledge base dedicated to Linux and applied mathematics.

Home > Mathematics > Derivative of a function > Derivative of square root of x

Derivative of square root of x

All the versions of this article: <English> <français> <italiano>

The derivative f’ of square root of x defined by f(x)=√x is for all x strictly positive f’(x)=1 / 2√x

Derivative of square root of x

The derivative $f’$ of the function $f(x)=\sqrt{x}$ is:

$$ \forall x \in ]0,+\infty[ ,\quad f’(x) = \frac{1}{2\sqrt{x}} $$

Proof/Demonstration

$$ \begin{aligned} \frac{df}{dx}=&\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}\\ =&\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \\ =&\lim _{h \rightarrow 0} \frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{h(\sqrt{x+h}+\sqrt{x})}\\ =&\lim _{h \rightarrow 0} \frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}\\ =&\lim _{h \rightarrow 0} \frac{h}{h(\sqrt{x+h}+\sqrt{x})}\\ =&\lim _{h \rightarrow 0} \frac{1}{\sqrt{x+h}+\sqrt{x}}=\frac{1}{2 \sqrt{x}} \end{aligned} $$

We conclude that:

$$ \forall x \in ]0,+\infty[ ,\quad f’(x) = \frac{1}{2\sqrt{x}} $$

Also in this section

  1. Chain rule proof - derivative of a composite function
  2. Derivative of 1/x
  3. Derivative of arccos x
  4. Derivative of arcsin x
  5. Derivative of arctan x
  6. Derivative of cos x
  7. Derivative of exp x, e^x
  8. Derivative of inverse functions
  9. Derivative of ln u
  10. Derivative of ln x
  11. Derivative of sin x
  12. Derivative of square root of x
  13. Derivative of tan x
  14. Derivative of x power n