Differentiation - Calculus - Mathematics Previous Year Questions

$$\text { If } y=\left|\begin{array}{ccc} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{array}\right| \text {, then } \frac{d y}{d x} \text { is equal to }$$

\(\sqrt[3]{y} \sqrt{x}=\sqrt[6]{(x+y)^5}\), then \(\frac{d y}{d x}=\)

If \(x=a \sec ^2 \theta\) & \(y=a \tan ^2 \theta\), then \(\frac{d^2 y}{d x^2}=\)

If \([x]\) represents the greatest integer function and \(f(x)=x-[x]-\cos x\), then \(f^{\prime}\left(\frac{\pi}{2}\right)=\)

If \(y=2 x^{n+1}+\frac{3}{x^n}\), then \(x^2 \frac{d^{2 y}}{d x^2}\) is

If \(2^x+2^y=2^{x+y}\), then \(\frac{d y}{d x}\) is

If \((x e)^y=e^y\), then \(\frac{d y}{d x}\) is

If \(y=\left(\cos x^2\right)^2\), then \(\frac{d y}{d x}\) is equal to

If \(a\) and \(b\) are fixed non-zero constants, then the derivative of \(\frac{a}{x^4}-\frac{b}{x^2}+\cos x\) is \(m a+n b-p\), where

Consider the following statements

Statement 1 : If \(y=\log _{10} x+\log _e x\), then \(\frac{d y}{d x}=\frac{\log _{10} e}{x}+\frac{1}{x}\)

Statement 2 : If \(\frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}\) and \(\frac{d}{d x}\left(\log _e x\right)=\frac{\log x}{\log e}\)

If the parametric equation of curve is given by \(x=\cos \theta+\log \tan \frac{\theta}{2}\) and \(y=\sin \theta\), then the points for which \(\frac{d y}{d x}=0\) are given by

For constant \(a, \frac{d}{d x}\left(x^x+x^a+a^x+a^a\right)\) is

If \(y=(x-1)^2(x-2)^3(x-3)^5\), then \(\frac{d y}{d x}\) at \(x=4\) is equal to