Chapter 5

Derivative of \(x^{6}+6^{x}\) with respect to \(x\) is (a) \(12 x\) (b) \(x+4\) (c) \(6 x^{5}+6^{x} \log 6\) (d) \(6 x^{5}+x 6^{x-1}\)

If \(y=\log _{10} x^{2}\),then \(\frac{d y}{d x}\) is equal to (a) \(2 / x\) (b) \(\frac{2}{x \log _{e} 10}\) (c) \(\frac{1}{x \log _{e} 10}\) (d) \(\frac{1}{10 x}\)

3\. If \(f(x)=\frac{1}{1-x}\) then derivative of composite function \(f[f\\{f(x)\\}]\) is \(\quad\) \mathrm{\\{} O r i s s a ~ J E E - 2 0 0 3 ] ~ (a) 0 (b) \(1 / 2\) (c) 1 (d) 2

Derivative of \(f(x)=\left|x^{2}-x\right|\) at \(x=2\) is (a) \(-3\) (b) 0 (c) 3 (d) Undefined

If \(y=e^{\left(1+\log _{c} x\right)}\), then \(\frac{d y}{d x}=\) [MPPET-1996; Pb.CET-2001] (a) \(e\) (b) 1 (c) 0 (d) \(\log _{e} x e^{\log \alpha x}\)

If \(y=e^{x} \log x\), then \(\frac{d y}{d x}\) is \(\quad\) ISCRA-1996] (a) \(\frac{e^{x}}{x}\) (b) \(e^{x}\left(\frac{1}{x}+x \log x\right)\) (c) \(e^{x}\left(\frac{1}{x}+\log x\right)\) (d) \(\frac{e^{x}}{\log x}\)

If \(f(x)=e^{x} g(x), g(0)=2, g^{\prime}(0)=1\), then \(f^{\prime}(0)\) is (a) 1 (b) 3 (c) 2 (d) 0

If \(y=\log _{10} x+\log _{x} 10+\log _{x} x+\log _{10} 10\), then \(\frac{d y}{d x}=\) (a) \(\frac{1}{x \log _{e} 10}-\frac{1}{\left(x \log _{e} x\right)^{2}}\) (b) \(\frac{1}{x \log _{e} 10}-\frac{1}{x \log _{10} e}\) (c) \(\frac{1}{x \log _{e} 10}-\frac{\log _{e}^{10}}{x\left(\log _{e} x\right)^{2}}\) (d) None of these

If \(y=f(x)\) is an even function such that \(f^{\prime}(0)\) exists, then \(f^{\prime}(0)=\) [IIT-1987] (a) 0 (b) \(-1\) (c) 1 (d) None of these

If \(f(x)\) an odd differentiable function defined on \((-\infty,+\infty)\) such that \(f^{\prime}(3)=2\), then \(f^{\prime}(-3)\) is [IIT-JEE1992]

The function \(f\) is differentiable with \(f(1)=\) 8 and \(f^{\prime}(1)=\frac{1}{8} .\) If \(f\) is invertible and \(g=f^{-1}\). Then \(g^{\prime}(8)=\) (a) 8 (b) \(1 / 8\) (c) 0 (d) 1

Given \(f(x+y+z)=f(x) f(y) f(z)\) for all \(x, y, z\) If \(f(2)=4\) and \(f^{\prime}(0)=3\), then \(f^{\prime}(2)=\) (a) 12 (b) \(-12\) (c) \(+12\) or \(-12\) (d) 1

The differential coefficient of the function \(|x-1|+|x-3|\) at the point \(x=2\) is \mathrm{\\{} [ R P E T - 2 0 0 2 ; ~ P b C E T - 2 0 0 0 , 0 4 ] ~ (a) \(-2\) (b) 0 (c) 2 (d) undefined

If \(f(x)=m x^{2}+n x+p\), Then \(f^{\prime}(1)+f^{\prime}(4)-f^{\prime}(5)\) is equal to [MPPET-2008] (a) \(m\) (b) \(-m\) (c) \(n\) (d) \(-n\)