The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue(marginal revenue).If the total revenue(in rupees)received from the sale of x units of a product is given by R9x)=3x²+36+5,find the marginal revenue,when=5,and write which value does the question indicate?

Differentiate w.r.t. x the function in Exercises \(x^{x^2-3}+(x-3)^{x^2}, \text { for } x>3\)

Find the points on the curve \({x^2\over4}+{y^2\over 25}=1\)at which the  tangents are
(i) parallel to the x-axis.
(ii)parallel to the y-axis.

For the function y=x³, if x=5 and \(\Delta \)x=0.01,find \(\Delta \)y

f(x)=-|x+1|+

The volume of a cube increasing at the rate of 9 cm³/sec. How fast is the surface area increasing when the length of an edge is 10 cm.

Find the point on the curve \(y^2 =8x + 3\)  for which the y-coordinate changes 4 times more than coordinate of x.

Show that the tangents to the curve \(y=x^2-7x+18\) at (3, 0) and (4, 0) are at right angles.

Use differentials find the approximate value of \((0.037)^{1/2}\)

Find an angle \(\theta\) which increases twice as fast as its sine.

The volume of a cube is increasing at the rate of 9 cubic centimetres per second. How fast is the surface area increasing when the length of an edge is 10 centimetres?

Show that the function  given by:
\(f(x)=7x-3\) is strictly increasing on R.

Find the equations of the tangent and normal to the curve \(\frac { x^{ 2 } }{ a^{ 2 } } =\frac { y^{ 2 } }{ b^{ 2 } } =1\) at the point \((\sqrt { 2a,b } )\)

Show that the height of the cylinder, open at the top, of given surface area and greatest volume is equal to the radius of its base.

Show that a cylinder of given volume open at the top has maximum total surface provided it height is equal to radius of its base.

Gas is escaping from a spherical balloon at the rate of 900cm³ /sec. How fast is the surface area,radius of balloon shrinking when the radius of the balloon is 30cm?

Radius of a variable circle is changing at the rate of 5cm/sec. What is the radius of the circle at the times when area is changing at the rate of 100cm² /sec?

\(f(x)={1\over{x}^{2}+1}\)

Find the increasing and decreasing function of \(f(x)=x+cos x \) in [0,2].