Chapter 1 - Differentiation
Exercise 1.2
Q.4
Q.4 )
$\begin{aligned}
f(x)&=x^{3}+x-2 \quad\left(f^{-1}\right)^{\prime}(0)\\
&Diff \quad w.r.t \quad x\\
\frac{d}{d x}f(x)&=\frac{d}{d x}(x^{3}+x-2)\\
f^{\prime}(x)&=3 x^{2}+1\\
\left(f^{-1}\right)^{\prime}&=\frac{1}{f^{\prime}(x)}\\
&=\frac{1}{3 x^{2}+1}\\
\left(f^{-1}\right)^{\prime}&=\frac{1}{3(0)^{2}+1}\\
&=\frac{1}{1}\\
&=1\end{aligned}$
Q.5
i)
$\begin{aligned}
&\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\\
Let \\
f(x)&=\tan ^{-1} x+\cot ^{-1} x \ldots \ldots (1) \\
&Diff \quad w.r.t \quad x\\
\frac{d}{d x} f(x)&=\frac{d}{d x}\left(\tan ^{-1} x+\cot ^{-1} x\right)\\
f^{\prime}(x)&=\frac{1}{1+x^{2}}+\frac{-1}{1+x^{2}}\\
f^{\prime}(x)&=\frac{1}{1+x^{2}}-\frac{1}{1+x^{2}}=0\\
f^{\prime}(x)&=0\\
&\therefore \text { f(x) is constant function}\\
Let,\\
f(x)&=k \quad \text {for any value at x}\\
f(0)&=\tan ^{-1}(0)+\cot ^{-1}(0)\\
&=0+\frac{\pi}{2} \\
\therefore k&=\frac{\pi}{2}=f(x)\end{aligned}$
ii)
$\begin{aligned}
&\sec ^{-1}(x)+\operatorname{cosec}^{-1}(x)=\frac{\pi}{2}\\
Let,\\
f(x)&=\sec ^{-1}(x)+\operatorname{cosec}^{-1}(x)\\
&Diff \quad w.r.t \quad x \\
\frac{d}{d x} f(x)&=\frac{d}{d x}\left[\sec ^{-1}(x)+\operatorname{cosec}^{-1}(x)\right]\\
f^{\prime}(x)&=\frac{1}{x \sqrt{x^{2}-1}}-\frac{1}{x \sqrt{x^{2}-1}}\\
f^{\prime}(x)&=0\\
\therefore f(x) &=\text { constant function } \\
f(x) &=k \quad \text {for any value of x}\\
f(1) &=\sec ^{-1}(x)+\operatorname{cosec}^{-1}(x) \\ &=\sec ^{-1}(1)+\operatorname{cosec}^{-1}(1) \\ &=0+\frac{\pi}{2}\\
&\therefore \sec ^{-1}(x)+\operatorname{cosec}^{-1}(x)=\frac{\pi}{2}\end{aligned}$
Exercise 1.2
