Explicit and Implicit Forms
A function is in explicit form if it is in function notation or the equation is solved for the
dependent variable, usually y.
– All of the examples we have seen so far have been in explicit form.
A function is in implicit form if it is written as an equation and is not solved for the dependent
variable.
– You may or may not be able to solve for the dependent variable.
– Even though you may not solve for y, y is assumed to be a “function” of x.
– Examples
1. xy = 1
2. x2 – 2x + y2 + 4y = 4
3. x = y3 – 3y2 + 2y
4. 3x2y3 – 2x/y + 7 = 0
Implicit Differentiation
Implicit differentiation is the process of taking the derivative of an implicit function that can’t be
solved for y.
To differentiate implicitly,
1. Differentiate each term with respect to x.
If the term involves just x, differentiate as usual.
• Examples
x3
sinx
x 1
x 2
If the term involves just y, differentiate as usual but then multiply by dy/dx or y’.
• This is an application of the chain rule, because y is assumed to be function of x.
• Examples
2y 1
y4
cosy
3y 2
– If the term involves both x and y, use the product or quotient rule and the above.
• Examples
x3
3 4
xy
sinxtany
cos y 2
2. Solve for dy/dx or y’.
Note
Everything we did with the explicit derivative applies to the implicit derivative as well.
• Slope of the tangent line.
– Now, there may be more than one tangent line for a particular x.
• Velocity and acceleration.
• Higher order derivatives.
– Generally, you want the derivatives to only involve x and y.
Examples
Find the derivative of
2x
+ 7xy = cosx.
y
Find the second derivative of 3x2y3 – 2x + 6y2 = 5.
Find the equation of the tangent lines of x = y3 – 3y2 + 2y at x = 0.
The position of an electron is given by x2 – 2x + y2 + 4y = 4. Find the velocity and acceleration
of the electron.
Applications
1. We can now prove the power rule for fractional exponents by rewriting the explicit function
as an implicit function with integer exponents.
– Power rule: If f(x) = xn, where n is a rational number, then f’(x) = nxn-1, if it exists.
2. We can now find the derivative of inverse functions.
1
– If f(x) = cos-1x = arccosx, then f ' ( x)
1 x2