Derivative of One Function w.r.t Another

Q.

If $u={\tan }^{-1}\left(\frac{x^3+y^3}{x-y}\right)$, then$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$is equal to

1. $-\sin 2u$

2. $\sin 2u$

3. $\cos 2u$

4. $-\cos 2u$

Q. if tan-1(x2-y2/x2+y2) = a, show that dy/dx= x(1- tan a)/y(1+ tan a)

Q. Let $f$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2).$ If $f\left(0\right)=0,$ $f\left(1\right)=1$ and $f\left(2\right)=2,$ then

1. $f^{\prime \prime }\left(x\right)>0$ for all $x\in (0,2)$
2. $f^{\prime }\left(x\right)=0$ for some $x\in [0,2]$
3. $f^{\prime \prime }\left(x\right)=0$ for some $x\in (0,2)$
4. $f^{\prime \prime }\left(x\right)=0$ for all $x\in (0,2)$

Q.

Let $y=y\left(x\right)$ be the solution of the differential equation, $\frac{2+\sin x}{(y+1)}\frac{dy}{dx}=-\cos x,y>0,y\left(0\right)=1.$ $Ify\left(\pi \right)=a$, and $\left(\frac{dy}{dx}\right)atx=\pi isb$ then the ordered pair $(a,b)$ is equal to

1. $\left(2,\frac{3}{2}\right)$

2. $(1,1)$

3. $(2,1)$

4. $(1,-1)$

Q.

If verify that A³ − 6A² + 9A − 4I = O and hence find A⁻¹

Q. 3. What is the value of sin inverse 5/13

Q.

If $\cos x=\tan y,coty=\tan z$ and $cotz=\tan x$, then $\sin x$ is equal to

1. $\frac{(\sqrt{5}+1)}{4}$

2. $\frac{(\sqrt{5}-1)}{4}$

3. $\frac{(\sqrt{5}+1)}{2}$

4. $\frac{(\sqrt{5}-1)}{2}$

Q. What is domain

Q.

If (x + iy)³ = u + iv, then show that.

Q. Let f(x)= [x] and g(x)=|x| then find (gof) (-1/3) - (fog) (1/3) .

Q.

Using Binomial Theorem, evaluate (99)⁵

Q.

Let f: R → R be defined as f(x) = 3x. Choose the correct answer.

(A) f is one-one onto (B) f is many-one onto

(C) f is one-one but not onto (D) f is neither one-one nor onto

Q. 59.The value of cos inverse (4/5) + tan inverse (3/5) is equal to (1) tan inverse (27/11) (2) sin inverse (11/27) (3) cos inverse (11/27) (4) zero

Q.

y= f(x)= ax-b/bx-a, show that x=f(y)

Q. Let $f:R\to R$ be a one-one onto differentiable function, such that f(2) = 1 and f (2) = 3. Then the value of ${\left(\frac{d}{\mathrm{dx}}\left(f^{-1}\left(x\right)\right)\right)}_{x=1}$ is

1. 1
2. $\frac{1}{2}$
3. $\frac{1}{3}$
4. 3

Q. 5. Find the value of p and q so that the given function is differentiable at x=1

Q. Let f:(-2, 2)-(-2, 2) be a continuous function such that f(x)=f(x^2) for every value of x belongs to df(domain) and f(0)=1/2, then the value of 4f(1/4) is equal to

Q.

Find the second order derivative of the given functions.

$ta{n}^{-1}x$

Q. Rational function is defined as f(x)/g(x). In this expression domain of f(x) is set of real numbers. Is domain of g(x) also set of real numbers except zero?

Q.

The function f(x) = 3x + 1 is

1. Insufficient information

2. injective

3. not injective

4. It is injective in some intervals of the domain

Q.

Let A = R − {3} and B = R − {1}. Consider the function f: A → B defined by

. Is f one-one and onto? Justify your answer.

Q.

Find the derivative of the following functions from first principle.

(i) x³ – 27 (ii) (x – 1) (x – 2)

(ii) (iv)

Q. Find the integral of the function $\frac{1}{x^2+a^2}$

1. ${\tan }^{-1}\left(\frac{x}{a}\right)$
2. $\frac{1}{a}{\tan }^{-1}\left(x\right)$
3. $\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)$
4. ${\tan }^{-1}\left(\frac{x}{a}\right)$

Q.

What type of function is the exponential function e^x function on R ?

1. Neither onto nor one-one

2. one-one and into

3. Many one

4. One-one and onto

Q. Let f(x) be a continuous function defined such that x is greater than equal to 1 but less than equal to 3. If f(x) takes rational value for all x and f(2)=5 then f(1.5) =k Find k

Q. Let $z=(\cos x{)}^5$ and $y=\sin x$. Then the value of $2\frac{d^{2}z}{d{y}^2}$ at $x=\frac{2\pi }{9}$ is

Q. 3. x- sin t, y- cos 2t

Q. Rate of change of $sinx$ with respect to $cosx$ at $x=\frac{\pi }{2}$ will be equal to

1. \N
2. 1
3. -1
4. $-\infty$

Q. A ={1, 2, 3, 4} find total bijective functions from A-A such that f(1)NOT1 f(2)=NOT/2 f(3)NOT=/3 F(4)=not4

Q. If y⁴+5xy+y=2 then find the value of dy/dx and d²y/dx².