A function can be explicit or implicit:
Explicit: y = f(x). In this case, f(x) is composed of one variable, x.
Examples include:
y = sin(x)
y = x2 + 3
y = ln(x) + 3x + 2
Implicit: In this form, the function is expressed in terms of both y and x.
Examples include:
25 = x2 + y2
4 = x3 + y4 + 2y + 3
y is treated as a function and must be differentiated with respect to x.
Example 1 – what is the derivative of y?
In the context of implicit differentiation, the derivative of y would be y’ or dy/dx.
Example 2 – what is the derivative of y2?
y is the inside function; (u)2 is the outside function. Applying the chain rule,
g’(x) * f’(g(x)) = y’ * 2(y)2 = y’*2y2
Steps:
1. differentiate both sides of the equations
2. collect all the y’ terms on one side
3. solve for y’
Example 3 – Differentiate 25 = x2 + y2
Differentiate both sides.
0 = 2x2-1 + y’ * 2(y)2 = 2x + y’*2y2
Collect the y’ terms on one side and solve for y’.
0-2x = 2x-2x+ y’*2y2
-2x = y’ * 2y2
-2x /2y2 = y’ * 2y2/2y2
-x/y2 = y’
Example 4 – Differentiate 4 = x3 + y4 + 2y + 3
Differentiate both sides.
0 = 3x2 + 4y3*y’ + 2y + 0
Collect the y’ terms on one side and solve for y’.
0-3x2-2y= 3x2-3x2+4y3*y’+2y-2y +0
-3x2-2y=4y3*y’
(-3x2-2y)/4y3=4y3/4y3*y’
(-3x2-2y)/4y3=y’
Leave A Comment