Question

Find the second derivative of the function ${{x}^{20}}$ using the derivative formula.

Answer 1

- Hint: First, we should know the first order derivative i.e. rate of change of y with respect to x represented as $\dfrac{dy}{dx}$. Formula will be $\dfrac{d}{dx}{{x}^{n}}=n\cdot {{x}^{n-1}}$ . Similarly, here we need to differentiate function $f\left( x \right)={{x}^{20}}$ by taking variable $y=f\left( x \right)$ and will be differentiating twice till we get $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ .

Complete step-by-step solution -

Now, the second order derivative means we have the function $f\left( x \right)$ which we will be differentiating one time with respect to x and we will be getting $\dfrac{dy}{dx}$ . Similarly, we will be repeating same thing in order to get our desired answer i.e. $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ .
We have the function $f\left( x \right)={{x}^{20}}$ . So, applying derivative formula which is $\dfrac{d}{dx}{{x}^{n}}=n\cdot {{x}^{n-1}}$
Therefore, taking variable $y=f\left( x \right)$
$y={{x}^{20}}$
Differentiating on both side with respect to x, we get
$\dfrac{dy}{dx}=\dfrac{d}{dx}\left( {{x}^{20}} \right)$
$\dfrac{dy}{dx}=20\cdot {{x}^{20-1}}$
$\dfrac{dy}{dx}=20\cdot {{x}^{19}}$ ………………………..(i)
Again, differentiating on both sides of equation (i) with respect to x which is known as second derivative.
$\dfrac{d}{dx}\left( \dfrac{dy}{dx} \right)=\dfrac{d}{dx}\left( 20\cdot {{x}^{19}} \right)$
$\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=20\cdot \dfrac{d}{dx}\left( {{x}^{19}} \right)$ (here, 20 can be taken outside as it is constant)
$\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=20\cdot \left( 19{{x}^{19-1}} \right)$
$\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=20\cdot \left( 19{{x}^{18}} \right)$
$\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=380{{x}^{18}}$
Thus, the second derivation of the function $f\left( x \right)={{x}^{20}}$ is $380{{x}^{18}}$.

Note: Students might get confused between integration and derivation as both have the same concept but different formulas. Integration formula is $\int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1}+c}$ and that of differentiation is $\dfrac{d}{dx}{{x}^{n}}=n\cdot {{x}^{n-1}}$ . Also, I should have a clear understanding where to use integration and where to use differentiation and should check the calculations errors.