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If X = f(t) and Y = g(t) Are Differentiable Functions of t , then prove that y is a differentiable function of x and
`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`
Hence find `"dy"/"dx"` if x = a cos2 t and y = a sin2 t.
Concept: Derivatives of Functions in Parametric Forms
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Concept: Second Order Derivative
If y = sin -1 `((8x)/(1 + 16x^2))`, find `(dy)/(dx)`
Concept: Derivatives of Functions in Parametric Forms
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`
Concept: Second Order Derivative
Evaluate : `int (sec^2 x)/(tan^2 x + 4)` dx
Concept: Derivatives of Functions in Parametric Forms
If `x^y = e^(x - y)` , show that `(dy)/(dx) = logx/(1 + logx)^2`
Concept: Derivative of Inverse Function
The total cost function of a firm is C = x2 + 75x + 1600 for output x. Find the output for which the average cost ls minimum. Is CA= Cm at this output?
Concept: Derivative of Inverse Function
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Concept: Increasing and Decreasing Functions
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Concept: Increasing and Decreasing Functions
Find `dy/dx,if e^x+e^y=e^(x-y)`
Concept: Increasing and Decreasing Functions
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Concept: Increasing and Decreasing Functions
Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.
Concept: Application of Derivatives to Economics
A manufacturing company produces x items at the total cost of Rs (180 + 4x). The demand function of this product is P = (240 − x). Find x for which profit is increasing.
Concept: Application of Derivatives to Economics
Find the elasticity of demand, if the marginal revenue is 50 and price is Rs 75.
Concept: Application of Derivatives to Economics
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Concept: Increasing and Decreasing Functions
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Concept: Increasing and Decreasing Functions
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Concept: Increasing and Decreasing Functions
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Concept: Maxima and Minima
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
Concept: Maxima and Minima
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
Concept: Maxima and Minima
