List of practice Questions

Differentiate the function with respect to \(x\). 
\((x+\frac{1}{x})^x+x^{(1+\frac{1}{x})}\)

Differentiate the function with respect to \(x\). 
\((x+3)^2.(x+4)^3.(x+5)^4\)

Differentiate the function with respect to \(x\). 
\(x^x-2^{sinx}\)

Differentiate the following w.r.t. \(x\):\((log\,x)^{cos\,x}\)

Differentiate the following w.r.t. \(x\):
\(\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\)

Differentiate the following w.r.t. \(x\):
\(cos(log\,x+e^x),x>0\)

Differentiate the following w.r.t. \(x\):
\(\frac{cos\,x}{log\,x},x>0\)

Differentiate the following w.r.t. \(x\):
\(log(log\,x),x>1\)

Differentiate the following w.r.t. \(x\):
\(\sqrt{e^{\sqrt{x}}},x>0\)

Differentiate the following w.r.t. \(x\):
\(e^x+e^{x^2}+....+e^{x^5}\)

Differentiate the following w.r.t. \(x\):
\(log(cos\,e^x)\)

Differentiate the following w.r.t. \(x\):
\(sin(tan^{-1}e^{-x})\)

A steel wire of length $l$ has a magnetic moment $M$. It is then bent into a semicircular arc. The new magnetic moment is

Find the intervals in which the function \(f\) given by \(f(x) = 2x^2 − 3x\) is (a) strictly increasing (b) strictly decreasing

Show that the function given by \(f(x) = sin x\) is (a) strictly increasing in \((0,\frac{π}{2})\) (b) strictly decreasing in \((\frac{π}{2},π)\) (c) neither increasing nor decreasing in \((0, π)\)

Show that the function given by \(f(x) = e^{2x}\) is strictly increasing on \(R\).

A balloon, which always remains spherical, has a variable diameter \(\frac{3}{2}(2x+1)\) Find the rate of change of its volume with respect to \(x\)

The radius of an air bubble is increasing at the rate of \(\frac{1}{2} cm/s\). At what rate is the volume of the bubble increasing when the radius is \(1 cm\)?

A particle moves along the curve \(6y=x^3+2\).Find the points on the curve at which the y coordinate is changing 8 times as fast as the \(x\) coordinate

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm

The length \(x\) of a rectangle is decreasing at the rate of \(5 cm/minute\) and the width \(y\) is increasing at the rate of \(4 cm/minute\). When \(x = 8 cm\) and \(y = 6 cm\), find the rates of change of \((a)\) the perimeter, and \((b)\) the area of the rectangle.

The radius of a circle is increasing at the rate of \(0.7 cm/s.\) What is the rate of increase of its circumference?

A stone is dropped into a quiet lake and waves move in circles at the speed of \(5 cm/s\). At the instant when the radius of the circular wave is \(8 cm\), how fast is the enclosed area increasing?

The radius of a circle is increasing uniformly at the rate of \(3 cm/s\). Find the rate at which the area of the circle is increasing when the radius is \(10 cm\).

The volume of a cube is increasing at the rate of \(8 cm^3 /s\). How fast is the surface area increasing when the length of an edge is \(12 cm\)?