University Maths Solution

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Maths Question
Question 1

$\text{Differentiate }y=\sin 3x\text{ from the first principle}$

Question 2

$\text{Differentiate }y=\sqrt{x}\text{ from the first principle}$

Question 3

$\text{Differentiate }y=\log x\text{ from the first principle}$

Question 4

$\text{Differentiate }y=\tan x\text{ from the first principle}$

Question 5

$\text{Differentiate }y=x\sec x\text{ from the first principle}$ 

Question 6

$\text{Differentiate }y=\text{ }x{{e}^{x}}\text{ from the first principle}$ 

Question 7

$\text{If }y=u+v\text{ where }y,u,v\text{ are functions of }x,\text{ show that }\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$

Question 8

$\begin{align}  & \text{Establish the formula for differentiating a quotient of two functions }y=\frac{u}{v}\text{ } \\ & \text{then }\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{{{v}^{2}}} \\\end{align}$

Question 9

$\text{Differentiate with respect to }x,\text{ }y=8{{x}^{4}}+7{{x}^{3}}+6{{x}^{2}}+5x+4$

Question 10

$\begin{align}  & \text{If }y=uvw\text{ } \\ & \text{show that }\frac{d}{dx}=u\frac{d}{dx}(vw)+v\frac{d}{dx}(uw)+w\frac{d}{dx}(uv) \\\end{align}$

Question 11

$\text{Differentiate with respect to }x,\text{ }y=4{{x}^{2}}\sin x{{\cos }^{2}}x$

Question 12

$\text{Differentiate with respect to }x,\text{ }y=3x\sin x\cos x$

Question 13

$\text{Differentiate }y={{({{x}^{2}}+1)}^{3}}\text{ with respect to }x$

Question 14

$\text{Differentiate }y=8{{x}^{2}}(1+\sin x)(1+\cos x)\text{ wrt }x$

Question 15

$\text{Differentiate }y=\frac{\cos x}{{{x}^{2}}+\sin x}\text{ with respect to }x$

Question 16

$\text{Differentiate }y=\frac{1}{1+\cos x}\text{ with respect to }x$

Question 17

$\text{Differentiate }y=\frac{x(x+1)}{(x+2)(x+3)}\text{ with respect to }x$

Question 18

$\text{Differentiate }y={{\sin }^{-1}}x\text{ with respect to }x$

Question 19

$\text{Differentiate }y={{\tan }^{-1}}\left( \frac{1+x}{1-x} \right)\text{ wrt }x$

Question 20

$\text{Differentiate }y={{x}^{x}}+{{e}^{\tan x}}\text{ with respect to }x$

Question 21

$\text{Differentiate }y={{x}^{\tfrac{2}{3}}}\text{ wrt }x$

Question 22

$\text{Differentiate }y={{\log }_{a}}{{x}^{2}}\text{ wrt }x$

Question 23

$\text{Differentiate }y=\sec \sqrt{x}\text{ wrt }x$

Question 24

$\text{Differentiate }y=x{{\sin }^{-1}}x\text{ wrt }x$

Question 25

$\text{Differentiate }y=x{{\log }_{e}}x-x\text{ wrt }x$

Question 26

$\text{Differentiate }y={{e}^{\sin x}}\text{ wrt }x$

Question 27

$\text{Differentiate }y=\sqrt{\frac{x-1}{x+1}}\text{ with respect to }x$

Question 28

$\text{If }y=A{{e}^{mx}}+B{{e}^{-mx}},\text{ where }A,B,\text{ and }M\text{ are constant Show that }\frac{{{d}^{2}}y}{d{{x}^{2}}}={{m}^{2}}y$

Question 29

$\text{If }y=\sin (m{{\sin }^{-1}}x)\text{ prove that }(1-{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}=x\frac{dy}{dx}-{{m}^{2}}y$

Question 30

$\text{If }x=a(\theta -\sin \theta )\text{ and }y=a(1-\cos \theta ),\text{ show that }1+{{\left( \frac{dy}{dx} \right)}^{2}}={{\operatorname{cosec}}^{2}}\frac{\theta }{2}$

Question 31

$\text{If }x=\frac{2+t}{1+2t}\text{ and }y=\frac{3+2t}{t}\text{ prove that }\frac{dy}{dx}=\frac{{{(1+2t)}^{2}}}{{{t}^{2}}}\text{ and find the value of }\frac{{{d}^{2}}y}{d{{x}^{2}}}\text{ when }x=0$

Question 32

$\text{Prove that }\frac{d}{dx}\left( \frac{1+\sin x+\cos x}{1-\sin x+\cos x} \right)=\frac{1}{1-\sin x}$

Question 33

$\text{If }y={{x}^{2}}\sin x,\text{ prove that  }{{x}^{2}}\frac{{{d}^{2}}y}{d{{x}^{2}}}-4x\frac{dy}{dx}+({{x}^{2}}+6)y=0$

Question 34

$\text{If }y={{\tan }^{-1}}x\text{ prove that }(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+2x\frac{dy}{dx}=0$

Question 35

$\text{Differentiate with respect to }x,\text{ }y={{\left( x-\frac{1}{x} \right)}^{-3}}$

Question 36

$\text{Differentiate with respect to }x,\text{ }y={{({{e}^{x}}\sin x)}^{2}}$

Question 37

$\text{If }y=\log (x+\sqrt{1+{{x}^{2}}})\text{ show that }(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}=0$

Question 38

$\text{Differentiate with respect to }x,\text{ }y={{\left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)}^{\tfrac{3}{2}}}$