What is the difference between implicit and explicit derivatives

Question

if sqrt (1-x^2)+ sqrt (1-y^2)=a(x-y) then show that dy/dx= sqrt(1-y^2)/sqrt(1-x^2)

Answer 1

Diffference between implicit and explicit differentiation :

An explicit function is one in which the function is in terms of the independent variable.

For explicit differentiation, the function is expressed in terms of independent variable and then differentiate to find derivative function.

Implicit functions are usually those functions in which terms of both dependent and independent variables.

For implicit differentiation, the function is applied derivative and then solved to find the derivative function.

Step 1:

Implicit Differentiation :

The function is

Let and

Substitute and in equation .

Trigonometric identity: then .

Trigonometric sum and difference property : .

Trigonometric sum and difference property : .

Substitute and in the above equation.

.

Step 2:

.

Differentiate implicitly on each side.

Derivative of a inverse trigonometric function : .

Therefore, the derivative function is .

Solution :

The derivative function is .

 

Answer 2

(2)

Step 1:

Explicit differentiation:

The function is .         (From (1))

Solve the function  in terms of .

Differentiate on each side with respect to .

Derivative of a inverse trigonometric function : .

Substitute and in the above function.

Trigonometric identity : then

If then .

.

Substitute in the .

Therefore, the derivative function is .

Solution :

The derivative function is .