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Second derivative of $$$x^{2}$$$

The calculator will find the second derivative of $$$x^{2}$$$, with steps shown.

Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator

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Your Input

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

Solution

Find the first derivative $$$\frac{d}{dx} \left(x^{2}\right)$$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$${\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = {\color{red}\left(2 x\right)}$$

Thus, $$$\frac{d}{dx} \left(x^{2}\right) = 2 x$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(x^{2}\right) = \frac{d}{dx} \left(2 x\right)$$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

Thus, $$$\frac{d}{dx} \left(2 x\right) = 2$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(x^{2}\right) = 2$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(x^{2}\right) = 2$$$A