1. Home
  2. Calculators
  3. Calculators: Calculus I
  4. Calculus Calculator

Second derivative of x\sqrt{x}

The calculator will find the second derivative of x\sqrt{x}, with steps shown.

Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator

Leave empty for autodetection.
Leave empty, if you don't need the derivative at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Find d2dx2(x)\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right).

Solution

Find the first derivative ddx(x)\frac{d}{dx} \left(\sqrt{x}\right)

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=12n = \frac{1}{2}:

(ddx(x))=(12x){\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right)\right)} = {\color{red}\left(\frac{1}{2 \sqrt{x}}\right)}

Thus, ddx(x)=12x\frac{d}{dx} \left(\sqrt{x}\right) = \frac{1}{2 \sqrt{x}}.

Next, d2dx2(x)=ddx(12x)\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right) = \frac{d}{dx} \left(\frac{1}{2 \sqrt{x}}\right)

Apply the constant multiple rule ddx(cf(x))=cddx(f(x))\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) with c=12c = \frac{1}{2} and f(x)=1xf{\left(x \right)} = \frac{1}{\sqrt{x}}:

(ddx(12x))=(ddx(1x)2){\color{red}\left(\frac{d}{dx} \left(\frac{1}{2 \sqrt{x}}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dx} \left(\frac{1}{\sqrt{x}}\right)}{2}\right)}

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=−12n = - \frac{1}{2}:

(ddx(1x))2=(−12x32)2\frac{{\color{red}\left(\frac{d}{dx} \left(\frac{1}{\sqrt{x}}\right)\right)}}{2} = \frac{{\color{red}\left(- \frac{1}{2 x^{\frac{3}{2}}}\right)}}{2}

Thus, ddx(12x)=−14x32\frac{d}{dx} \left(\frac{1}{2 \sqrt{x}}\right) = - \frac{1}{4 x^{\frac{3}{2}}}.

Therefore, d2dx2(x)=−14x32\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right) = - \frac{1}{4 x^{\frac{3}{2}}}.

Answer

d2dx2(x)=−14x32\frac{d^{2}}{dx^{2}} \left(\sqrt{x}\right) = - \frac{1}{4 x^{\frac{3}{2}}}A