Implicit Differentiation Example Problems :
Here we are going to see some example problems involving implicit differentiation.
A function in which the dependent variable is expressed solely in terms of the independent variable x, namely, y = f(x), is said to be an explicit function. For instance, y = (1/2)x3 - 1 is an explicit function, whereas an equivalent equation 2y − x3 + 2 = 0 is said to define the function implicitly or y is an implicit function of x.
Question 1 :
Find the derivatives of the following
√(x2 + y2) = tan-1(y/x)
Solution :
√(x2 + y2) = tan-1(y/x)
By finding derivation on left hand side, we get
= (1/2√(x2+y2))(2x+2y(dy/dx))
= [x+y(dy/dx)]/√(x2+y2) ----(1)
By finding derivation on right left hand side, we get
= 1/[1 + (y/x)2] (-y/x2) + (1/x) (dy/dx)
= x2/(x2 + y2) (x (dy/dx) - y)/x2
= (x (dy/dx) - y)/(x2 + y2) ----(2)
(1) = (2)
By equating (1) and (2), we get
[x+y(dy/dx)]/√(x2+y2) = (x (dy/dx) - y)/(x2 + y2)
√(x2+y2)[x+y(dy/dx)] = (x (dy/dx) - y)
x√(x2+y2) + y √(x2+y2)(dy/dx) - x (dy/dx) = -y
(dy/dx) (y√(x2+y2) - x) = -y-x√(x2+y2)
(dy/dx) = (x√(x2+y2) + y) / (x - y√(x2+y2))
Question 2 :
Differentiate the following
tan (x + y) + tan (x - y) = x
Solution :
tan (x + y) + tan (x - y) = x
Differentiate with respect to "x"
sec2(x + y) (1 + dy/dx) + sec2(x - y) (1 - dy/dx) = 1
sec2(x+y)+sec2(x+y)(dy/dx)=1-sec2(x-y)-sec2(x-y)(dy/dx)
sec2(x+y)(dy/dx)+sec2(x-y)(dy/dx) = 1-sec2(x-y)-sec2(x-y)
(dy/dx) (sec2(x+y)+sec2(x-y)) = 1 - sec2(x-y)- sec2(x-y)
dy/dx = (1 - sec2(x-y)- sec2(x-y)) / (sec2(x+y)+sec2(x-y))
Question 3 :
Differentiate the following
If cos (xy) = x, show that dy/dx = -(1+ysin(xy))/x sin(xy)
Solution :
cos (xy) = x
Differentiate with respect to "x"
-sin (xy) [x(dy/dx) + y(1)] = 1
-sin (xy)(dy/dx) - y sin (xy) = 1
-sin (xy)(dy/dx) = 1 + ysin(xy)
dy/dx = -[1 + ysin(xy)]/sin (xy)
Question 4 :
Differentiate the following
tan-1[√(1 - cos x)/(1+cosx)]
Solution :
Let y = tan-1[√(1 - cos x)/(1+cosx)]
Trigonometric formula for (1-cos x)/(1+cosx) = tan2(x/2)
By applying trigonometric formula, we may find derivatives easily.
y = tan-1[√tan2(x/2)]
y = x/2
Differentiating with respect to "x"
dy/dx = 1/2
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