Derivative of Some Standard Functions

Q. If limit x tends to zero (sin2x+asinx/x^3) = b (finite) , then ab equals

Q. d/dx of 1+tanx/1-tanx

Q. if x belongs to (pi\2, 3pi\2)value of tan^-1(tanX)

Q. 36. Find the derivative of sin square X with first principal

Q. If cos^-1(y)=sin^-1(x) , then locus of (x, y) is

Q. If z=re^{iθ, then prove that e^{iz\vert=e^{-r\operatorname{sinθ

Q. the value of 2 cot(cot^(-1)(3)+cot^(-1)(7)+cot^(-1)(13)+ cot^(-1)(21)) is

Q. Prove that:

${\cos }^2\left(\frac{\pi }{4}-x\right)-{\sin }^2\left(\frac{\pi }{4}-x\right)=\sin 2x$

Q. Let tan^(-1)x+tan^(-1)y=a+b tan^(-1)((x+y)/(1-xy)) then find the sum of all possible distinct values of a and b .

Q. cOs(n+1) a.cos (n-1) a- sin (n + 1) a.sin (n -1) a =

Q. \operatorname{sin}^2θ+\operatorname{cos}^4θ=\operatorname{cos}^2θ+\operatorname{sin}^4θ

Q.

Given h(x) = 2sinx and g(x) = cosx, then h . g is ?

1. 2sin²x

2. cos 2x

3. sin 2x

4. 2cos²x

Q.

Let $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}$ and $f^1\left(0\right)$ and $f\left(0\right)=b$ then $f^1\left(x\right)$ is equal to

1. 0

2. a

3. b

4. a+b

Q. 10. if f(x) = x + tan x , and g(x) is inverse of f(x), then g(x) is

Q. What is the second derivative of (sin^2 ax+b)?

Q. If f(x) = |x|, then f’(x), where x $\ne$ 0 is equal to

1. $-1$
2. $0$
3. $1$
4. $\frac{|x|}{x}$

Q.

Match the following
$\begin{array}{cc}Functions& Derivatives\\ \left(a\right)x^n& 1\left)\frac{1}{x}\right(lo{g}_{a}e)\\ \left(b\right)e^x& 2)\frac{1}{x},x>0\\ \left(c\right)a^x& 3)n{x}^{n-1},nisaconstant\\ \left(d\right)lo{g}_{e}x& 4)a^{x}lo{g}_{e}a,a>0\\ \left(e\right)lo{g}_{a}x& 5)e^x\end{array}$

1. a – 1, b – 2, c – 3, d – 4, e – 5

2. a – 1, b – 5, c – 4, d – 2, e – 3

3. a – 3, b – 5, c – 4, d – 2, e – 1

4. None of the above

Q. Find the principal value of cosec x=-1

Q.

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

Q. THE NUMBER OF SOLUTION(S) OF SIN^-1(X)+COS^-1(1-X)=SIN^-1(-X) IS /ARE

Q. Evaluate : sin 7pi/12 . cos pi/4 - cos 7pi/12 . sin pi/4

Q.

Match the following
$\begin{array}{cc}Functions& Derivatives\\ \left(a\right)x^n& 1\left)\frac{1}{x}\right(lo{g}_{a}e)\\ \left(b\right)e^x& 2)\frac{1}{x},x>0\\ \left(c\right)a^x& 3)n{x}^{n-1},nisaconstant\\ \left(d\right)lo{g}_{e}x& 4)a^{x}lo{g}_{e}a,a>0\\ \left(e\right)lo{g}_{a}x& 5)e^x\end{array}$

1. a – 1, b – 2, c – 3, d – 4, e – 5

2. a – 1, b – 5, c – 4, d – 2, e – 3

3. a – 3, b – 5, c – 4, d – 2, e – 1

4. None of the above

Q. Match the following

$\begin{array}{cc}Functions& Derivatives\\ \left(a\right)\sin x& 1)-\sin x\\ \left(b\right)\cos x& 2){\sec }^{2}x\\ \left(c\right)\tan x& 3)\cos x\\ \left(d\right)\sec x& 4)-cose{c}^{2}x\\ \left(e\right)\cot x& 5)\sec x\tan x\\ \left(f\right)cosecx& 6)-cosecxcotx\end{array}$

1. a – 1, b – 3, c – 2, d – 4, e – 5, f – 6
2. a – 3, b – 1, c – 2, d – 5, e – 4, f – 6
3. a – 2, b – 1, c – 3, d – 5, e – 4, f – 6
4. None of the above

Q. The value of ${\cos }^2\left(\frac{\mathrm{\pi }}{6}+x\right)-{\sin }^2\left(\frac{\mathrm{\pi }}{6}-x\right)$ is
(a) $\frac{1}{2}\cos 2x$
(b) 0
(c) $-\frac{1}{2}\cos 2x$
(d) $\frac{1}{2}$

Q. 55 If a=cosx+i sinx, find the value of (1+a)/(1-a)

Q. find domain of f(x) = √log_3(cos(sinx))