H.S.C.
Differentiation EX 1.1 Q.1 Mathematics and statistic for arts and science
Maharashtra state board Mathematics arts and Science part 2
Differentiation
i)
Tags
Differentiation
Exercise 1.1
Solution Ex 1.1
Q.1)i)
$$
\begin{aligned}
&\text { 1) } y=\left(x^{3}-2 x-1\right)^{5}\\
&\begin{array}{l}
\text { Diff w. } x \text { t. } x \\
\frac{d y}{d x} y=\frac{d}{d x}\left(x^{3}-2 x-1\right)^{5}\end{array} \\
&\frac{d y}{d x}=5\left(x^{3}-2 x-1\right)^{4} \cdot \frac{d}{d x}\left(x^{3}-2 x-1\right)\\
&=5\left(x^{3}-2 x-1\right)^{4}\left[\frac{d}{d x} x^{3}-\frac{d}{d x}(2 x)-\frac{d}{d x}(1)\right]\\
&=5\left(x^{3}-2 x-1\right)^{4}\left(3 x^{2}-2-0\right)\\
&=5\left(3 x^{2}-2\right)\left(x^{3}-2 x-1\right)^{4}
\end{aligned}
$$
ii) $$y=\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{\frac{5}{2}}\\
Diff w.r t x\\
\frac{d y}{d x}=\frac{d}{d x}\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{5 / 2}\\
=\frac{5}{2}\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{\frac{5}{2}-1} \cdot \frac{d}{d x}\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)\\
=\frac{5}{2}\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{3 / 2}\left(\frac{d}{d x} 2 x^{3 / 2}-\frac{d}{d x} 3 \cdot x^{4 / 3}-\frac{d}{d x} 5\right)\\
=\frac{5}{2}\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{\frac{3}{2}} \cdot\left(2 . \frac{3}{2} x^{\frac{3}{2}-1}-3.\frac{4}{3} \cdot x^{\frac{4}{3}-1}-0\right)\\
=\frac{5}{2}\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{\frac{3}{2}} \cdot\left(3 x^{\frac{1}{2}} \cdot 4 x^{\frac{1}{3}}\right)\\
=\frac{5}{2}(3 \sqrt{x}-4 \sqrt[3]{x})\left(2 x^{3 / 2}-3 x^{4 / 3}-5\right)^{3 / 2}\\$$
iii) $$y=\sqrt{x^{2}+4 x-7} \quad\left[\sqrt{x}=\frac{1}{2 \sqrt{x}}\right.\\
Diff. \omega \cdot r \cdot t \cdot x\\
\frac{d y}{d x}=\frac{1}{2 \sqrt{x^{2}+4 x-7}} \cdot \frac{d}{d x}\left(x^{2}+4 x-7\right)\\
=\frac{1}{2 \sqrt{x^{2}+4 x-7}}\left(\frac{d}{d x} x^{2}+\frac{d}{d x} 4 x-\frac{d}{d x} 7\right)\\
=\frac{1}{2 \sqrt{x^{2}+4 x-7}} \cdot(2 x+1-0)\\
=\frac{2(x+2)}{2 \sqrt{x^{2}+4 x-7}}\\
=\frac{(x+2)}{\sqrt{x^{2}+4 x-7}}\\$$
iv)$$y=\sqrt{x^{2}+\sqrt{x^{2}+1}}\\
Diff wrt \quad x\\
\frac{d y}{d x}=\frac{d}{d x}(\sqrt{x^{2}+\sqrt{x^{2}+1}})\\
=\frac{1}{2 \sqrt{x^{2}+\sqrt{x^{2}+1}}} \cdot \frac{d}{d x}\left(x^{2}+\sqrt{x^{2}+1}\right)\\
=\frac{1}{2 \sqrt{x^{2}+\sqrt{x^{2}+1}}} \cdot\left[\frac{d}{d x} x^{2}+\frac{d}{d x} \sqrt{x^{2}+1}\right]\\
=\frac{1}{2 \sqrt{x^{2}+\sqrt{x^{2}+1}}} \cdot\left[2 x+\frac{1}{2 \sqrt{x^{2}+1}} \frac{d}{d x}\left(x^{2}+1\right)\right]\\
=\frac{1}{2 \sqrt{x^{2}+\sqrt{x^{2}+1}}}\left[2 x+\frac{1}{2\sqrt{x^{2}+1}} 2x\right]\\
=\frac{1}{2 \sqrt{x^{2}+\sqrt{x^{2}+1}}} \cdot\left[2 x+\frac{x}{\sqrt{x^{2}+1}}\right]\\$$
v) $$y=\frac{3}{\quad 5 \sqrt[3]{2 x^{2}-7 x-5})^{5}}\\
y=\frac{3}{5\left(2 x^{2}-7 x-5\right)^{5 / 3}}\\
y=\frac{3}{5}\left(2 x^{2}-7 x-5\right)^{-5 / 3}\\
Diff w.r.t.\quad x\\
\frac{d y}{d x}=\frac{3}{5} \cdot \frac{d}{d x}\left(2 x^{2}-7 x-5\right)^{-\frac{5}{3}}\\
=\frac{3}{5} \left(\frac{-5}{3}\right) \cdot\left(2 x^{2}-7 x-5\right)^{-\frac{5}{3}-1} \times\frac{d}{d x}\left(2 x^{2}-7 x-5\right)\\
=(-1)\left(2 x^{2}-7 x-5\right)^{-\frac{8}{3}}\times\left(\frac{d}{d x} 2 x^{2}-\frac{d}{d x}(7 x)-\frac{d}{d x} 5\right)\\
=(-1)\left(2 x^{2}-7 x-5\right)^{-8 / 3}(4 x-7)\\
=\frac{-(4 x-7)}{\left(2 x^{2}-7 x-5\right)^{8 / 3}}\\$$
vi)$$y=\left[\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right]^{5}\\
Diff wrt x\\
\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{5}\\
=5 \cdot\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4} \cdot \frac{d}{d x}\left[\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right]\\
=5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\left[\frac{d}{d x} \sqrt{3 x-5}-\frac{d}{d x}(3 x-5)^\frac{-1}{2}\right]\\
=5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\left[\frac{1}{2 \sqrt{3 x-5}} \times \frac{d}{d x}(3 x-5)+\right.\frac{1}{2}(3 x-5)^{-\frac{1}{2}-1})\times\frac{d}{d x}(3 x-5)]\\
=5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\times\left[\frac{3}{2 \sqrt{3 x-5}} +\frac{(3 x-3)^{-3 / 2}}{2}.3\right]\\
=5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\times\left[\frac{3}{2 \sqrt{3 x-5}}+\frac{3}{2 \sqrt{(3 x-1)^{3}}}\right]\\
=5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\left[\frac{3}{2 \sqrt{3 x-5}}\left(1+\frac{1}{\sqrt{(3 x-5)^{2}})}\right)\right]\\
=5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\left[\frac{3}{2 \sqrt{3 x-5}}\left(1+\frac{1}{{(3 x-5)}}\right)\right]\\
5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4} \times \frac{3}{2 \sqrt{3 x-5}}\left(\frac{3 x-5+1}{3 x-5}\right)\\
5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}+\frac{3(3 x-4)}{2(3 x-5)^{1 / 2}(3 x-5)}\\
5\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4} \times \frac{3(3 x-4)}{2(3 x-5)^{3 / 2}}\\
\frac{15(3 x-4)}{{2(3 x-5)^{3 / 2}}}\left(\sqrt{3 x-5}-\frac{1}{\sqrt{3 x-5}}\right)^{4}\\$$
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