# CPT Chapter Differential Calculus

With Anand.Duration:4hrs 42 mins

What is Calculus?
Arithmetic is a study of quantity
Geometry study shapes
Trigonometry study traingle
Calculus studies change

Branches of Calculus-
Differentiation or Derivatives
Integration

Differentiation-
What is Differentiation?
Simple Formula
Product & Quotient Rule
Chain Rule
Logarithmic Expression
Parametric Function
Implicit Function
Second derivatives

Simple Formulaxn

ex

ax

xx

Examples-

If f(x)=xk and f '(1)=10, then the value of k is :
(a) 10
(b) -10
(c) 1/10
(d) None

If xy = 1 then y2 + dy/dx is equal to
a) 1
b) 0
c) –1
d) none of these

The slope of the tangent to the curve y = x2 –x at the point, where the line y = 2 cuts the curve in the First quadrant, is
a) 2 b) 3 c) –3 d) none of these

For the curve x2 + y2 + 2gx + 2hy = 0, the value of dy/dx at (0, 0) is
a) -g/h
b) g/h
c) h/g
d) none of these

Product & Quotient Rule-

If x = y log (xy), then dy/dx equal to:
(a) x +y/x (1+logxy)
(b) x - y/x (1+logxy)
(c) x +y/x (logx+logy)
(d) x - y/x (logx+logy)

If xy (x-y)=0, find dy/dx :
(a) y(2x - y)/x(2y - x)
(b) x(2x - y)/y(2y - x)
(c) y(2y - x) /x(2x - y)
(d) None of these

The slope of the tangent at the point (2,-2) to the curve
x2+ xy+y2-4 = 0 is given by :
(a) 0
(b) 1
(c) - 1
(d) None

Chain Rule-

If x3y2=(x-y)5.Find dy/dx at (1,2).
(a)-7/9
(b) 7/9
(c) 9/7
(d) - 9/7

If y=log(5-4x2/3+5x2) , then dy = dx
(a) 8/(4x-5)-10/(3+5x)
(b) (4x2-5)-(3+5x2)
(c) -8x/(4x2-5)-10x/(3+5x2)
(d) 8x-10

If (x2/a2) - (y2/a2) = 1, (dy/dx) can be expressed as
a) x/a
b)x/√(x2-a2)
c){1/(x2/a2)-1}
d) none of these

If log (x / y) = x + y, (dy/dx) may be found to be
a)y(1-x)/x(1+y)
b)y/x
c)(1-x)/(1+y)
d)none of these

Log Functions-
Rules of Log-

Parametric Functions
y= 15t , x=log(t3+t)

y= 5t4+3 , x=4t

x=2t+5and y=t2-5,then dx/dy=?
(a)t
(b)-1/t
(c)1/t
(d)0

If x=ct, y=c/t, then dy/dx is equal to:
(a) 1/t
(b) t.et
(c) -1/t2
(d) None of these

Implicit Functions-

xy+y2x+15=0

If xy=yx, then dy/dx gives :
(a) x(x logy-y)/ y(ylogx-x)
(b) x(y Iogx-x)/ y(xlogy-y)
(c) y(x logy- y)/ x(ylogx-x)
(d) None of these

Second Derivative-

Find the second derivative of y=√x+1
(a) 1/2 (x +1)-1/2
(b) -1/4 (x+1)-3/2
(c) 1/4 (x +1)-1/2
(d) None of these

If y=2x+4/x, then x2 d2y/dx2+xdy/dx - y yields
(a) 3
(b) 1
(c) 0
(d) 4

Conclusion-

What is Differentiation?
Simple Formula
Product & Quotient Rule
Chain Rule
Logarithmic Expression
Parametric Function
Implicit Function
Second derivatives

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