Using mathematical induction prove that \(\frac{d}{dx}\)(xn)=nxn-1 for all positive integers n
If f(x)=|x|3, show that f"(x)exists for all real x, and find it.
For the curve \(y = 4x^3 − 2x^5\) , find all the points at which the tangents passes through the origin.
If x=a(cost+tsint) and y=a(sint-tcost),find \(\frac{d^2y}{dx^2}\)
If cosy=xcos(a+y) with cosa≠±1,prove that \(\frac{dy}{dx}\)=\(\frac{cos^2(a+y)}{sin\,a}\)
Find the least value of a such that the function f given \(f(x)=x^2+ax+1\) is strictly increasing on \((1, 2)\).
Prove that the function f given by \(f(x) = x^2 − x + 1\) is neither strictly increasing nor strictly decreasing,on \((−1, 1)\).
\(x\sqrt{1+y}+y\sqrt{1+x}=0\), for -1<x<1,prove that \(\frac{dy}{dx}\)=\(-\)\(\frac{1}{(1+x)^2}\)
Prove that the logarithmic function is strictly increasing on \((0, ∞)\).
Prove that \(y=\frac{ 4sinθ}{(2+cosθ)}-θ \)is an increasing function of \(θ\) in \([0,\frac π2]\).
Find \(\frac{dy}{dx}\), if y=sin-1x+sin-1\(\sqrt{1-x^2}\), -1≤x≤1
Find the values of x for which \(y=[x(x-2)]^2\) is an increasing function.
Find the general solution: \(\frac {dy}{dx}+2y=sin\ x\)
Show that \(y = log(1+x) - \frac {2x}{2+x}, \ x>-1\),is an increasing function of x throughout its domain.
The general solution of the differential equation \(e^{x}dy+(ye^{x}+2x)dx=0\) is
The general solution of a differential equation of the type \(\frac{dx}{dy}+p_{1}x=Q1\) is
Choose the correct answer.If x,y,z are nonzero real numbers,then the inverse of matrixA=\(\begin{bmatrix}x& 0& 0\\ 0& y& 0\\0&0& z\end{bmatrix}\)is
Choose the correct answer.Let A=\(\begin{bmatrix}1&sin\theta&1\\-sin\theta&1&sin\theta\\-1&-sin\theta&1\end{bmatrix}\),\(where 0≤\theta≤2\pi,then\)