H.S.C.

Differentiation EX 1.2 Q.2 Maharashtra state Board Mathematics and statistic for arts and science

11:09 PM
Chapter 1 - Differentiation 

 Exercise 1.2

 Q.2


i)
$\begin{aligned} y&=x^{2} \cdot e^{x}\\ & Diff \quad w.r.t \quad x \\ \frac{d y}{d x}&=\frac{d}{d x}\left(x^{2} \cdot e^{x}\right)\\ &=x^{2} \cdot \frac{d}{d x} e^{x}+e^{x} \cdot \frac{d}{d x} x^{2}\\ \frac{d y}{d x}&=e^{x} \cdot x^{2}+2 x \cdot e^{x}\\ &=x e^{x}(x+2) \\ \frac{d x}{d x} &=\frac{1}{\left(\frac{d y}{d x}\right)} \\ &=\frac{1}{x e^{x}(x+2)} \end{aligned}$
ii)
$\begin{aligned} y&=x \cdot \cos x\\ &Diff \quad w.r.t \quad x \\ \frac{d y}{d x}&=\frac{d}{d x}(x \cdot \cos x)\\ &=x \cdot \frac{d}{d x} \cos x+\cos x \cdot \frac{d}{d x} \cdot(x)\\ &=-x \sin x+\cos x\\ NOW,\\ \frac{d x}{d y} &=\frac{1}{\left(\frac{d y}{d x}\right)} \\ &=\frac{1}{-x \sin x+\cos x}\\ &=\frac{1}{\cos x-x \sin x}\end{aligned}$
iii)
$\begin{aligned} y&=x \cdot 7^{x}\\ & Diff\quad w.r.t. \quad x \\ \frac{d y}{d x}&=\frac{d}{d x}\left(x \cdot 7^{x}\right)\\ &=x \cdot \frac{d}{d x} 7^{x}+7^{x} \cdot \frac{d}{d x}(x)\\ &=x \cdot 7^{x} \cdot \log {7}+7^{x}\\ &=7^{x}(x\log 7+1)\\ Now,\\ \frac{d x}{d y}&=\frac{1}{\left(\frac{d y}{d x}\right)}\\ &=\frac{1}{7^{x}(x \log 7+1)}\end{aligned}$
iv)
$\begin{aligned} y&=x^{2}+\log x\\ &Diff \quad w.r.t.t \quad x \\ \frac{d y}{d x} &=\frac{d}{d x}\left(x^{2}+\log x\right) \\ &=2 x+\frac{1}{x} \\ &=\frac{2 x^{2}+1}{x} \\ Now, \\ \\ \frac{d x}{d y} &=\frac{1}{\left(\frac{d y}{d x}\right)} \\ &=\frac{1}{\frac{\left(2 x^{2}+1\right)}{x}} \\ &=\frac{x}{2 x^{2}+1} \end{aligned}$
V)
$\begin{aligned} y&=x \log x\\ &Diff \quad w .r.t \quad x \\ \frac{d y}{d x}&=\frac{d}{d x}(x \cdot \log x)\\ &=x \cdot \frac{d}{d x} \log x+\log x \cdot \frac{d}{d x} x \\ &=x \cdot \frac{1}{x}+\log x \\ \frac{d y}{d x} &=1+\log x \\\\ Now,\\ \frac{d x}{d y}&=\frac{1}{\left(\frac{d y}{d x}\right)}\\ &=\frac{1}{1+\log x} \end{aligned}$


Exercise 1.2

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