Additional Practice Questions Set 2:

What is the value of \(c\) such that the line \(y=2x-3\) is tangent to the parabola \(y=cx^2\)?  solution

Find the derivatives of the following functions. You do not need to simplify your answer. solution

(a) \(f(x)=(1+\tan^{-1}x)^3 \)   (b) \( f(x)=\sqrt{\dfrac{x^2-5}{3x+2}}\)   (c) \(f(x)=\sin^2(3x) \)   (d) \(f(x)=\ln(4x^3+x) \)

Find the derivatives of the following functions. You do not need to simplify your answer.  solution

(a) \(f(x)= e^{5x}+x^5+5^x\)   (b) \( f(x)=\ln(x \tan x) \)   (c) \(f(x)=\left(\dfrac{x^3+2x}{5x^2-3} \right)^7\)

Find the derivatives of the following functions. You do not need to simplify your answer.   solution

(a) \(f(x)=(\sin^{-1}x-x^3)^2 \)   (b) \( f(x)=\sin^2 x + \cos^2 x \)  (c) \(f(x)=2^{3x^2-5x+\pi}\)  

Find the derivatives of the following functions. You do not need to simplify your answer.   solution

(a) \(f(x)=(\cos^{-1}x+x^2)^2 \)    (b) \( f(x)=\dfrac{\tan x}{1-\sec x}\)    (c) \(f(x)=e^{x^3\ln x}\)   (d) \(f(x)=7^{4x^3+x} \)

Find the slope of the tangent line to the curve \( \tan(xy) = y \) at the point \( \left(\frac{\pi}{4}, 1 \right)\). solution

Find an equation of the tangent line to the curve \( y=\sin(\tan x) \) at \(x=0\).   solution