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Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation2 Marks
Question
Find the $n^{\text {th }}$ derivative of the following : $x^m$
✓

Answer

Let $y=x^m$
Differentiate w.r.t. $x$
$
\frac{d y}{d x}=\frac{d}{d x}\left(x^m\right)=m x^{m-1}
$
Differentiate w.r.t. $x$
$
\begin{aligned}
& \frac{d}{d x}\left(\frac{d y}{d x}\right)=m \frac{d}{d x} x^{m-1} \\
& \frac{d^2 y}{d x^2}=m \cdot(m-1) x^{m-2}
\end{aligned}
$
Differentiate $w \cdot r \cdot t \cdot x$
$
\begin{aligned}
& \frac{d}{d x}\left(\frac{d^2 y}{d x^2}\right)=m \cdot(m-1) \frac{d}{d x}\left(x^{m-2}\right) \\
& \frac{d^3 y}{d x^3}=m \cdot(m-1) \cdot(m-2) x^{m-3}
\end{aligned}
$
In general $n^{\text {th }}$ order derivative will be
$
\begin{aligned}
& \frac{d^n y}{d x^n}=m \cdot(m-1) \cdot(m-2) \ldots[m-(n-1)] x^{m-n} \\
& \frac{d^n y}{d x^n}=m \cdot(m-1) \cdot(m-2) \ldots[m-n+1] x^{m-n}
\end{aligned}
$

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