The chain rule is an important piece of knowledge to have when dealing with calculus problems (including implicit differentiation problems). The chain rule states that for a function F(x) which can be written as (f o g)(x), the derivative of F(x) is equal to f'(g(x))g'(x). What is Implicit Differentiation? 🎥Watch: AP Calculus AB/BC - Implicit Derivatives. Back in pre-calculus, you likely learned or talked about how there are two different types of equations: explicit equations and implicit equations. An explicit equation is written such that y is isolated on one side. For example, y=2x+3 is an explicit equation. Using a calculator, the value of to four decimal places is 3.0166. The value given by the linear approximation, 3.0167, is very close to the value obtained with a calculator, so it appears that using this linear approximation is a good way to estimate, at least for near 9.
6.1 Implicit Vs Explicitap Calculus 2nd Edition
Finding the derivative when you can't solve for y
You may like to read Introduction to Derivatives and Derivative Rules first.
Implicit vs Explicit
A function can be explicit or implicit:
Explicit: 'y = some function of x'. When we know x we can calculate y directly.
Implicit: 'some function of y and x equals something else'. Knowing x does not lead directly to y.
Example: A Circle
Explicit Form | Implicit Form |
y = ± √ (r2 − x2) | x2 + y2 = r2 |
In this form, y is expressed as a function of x. | In this form, the function is expressed in terms of both y and x. |
The graph of x2 + y2 = 32
How to do Implicit Differentiation
- Differentiate with respect to x
- Collect all the dydx on one side
- Solve for dydx
Example: x2 + y2 = r2
Differentiate with respect to x:
ddx(x2) + ddx(y2) = ddx(r2)
Let's solve each term:
Which gives us:
2x + 2ydydx = 0
Collect all the dydx on one side
ydydx = −x
Solve for dydx:
dydx = −xy
The Chain Rule Using dydx
Let's look more closely at how ddx(y2) becomes 2ydydx
The Chain Rule says:
dudx = dudydydx
Substitute in u = y2:
ddx(y2) = ddy(y2)dydx
And then:
ddx(y2) = 2ydydx
Basically, all we did was differentiate with respect to y and multiply by dydx
Another common notation is to use ' to mean ddx
The Chain Rule Using '
The Chain Rule can also be written using ' notation:
f(g(x))' = f'(g(x))g'(x)
g(x) is our function 'y', so:
f(y)' = f'(y)y'
f(y) = y2, so f'(y) = 2y:
f(y)' = 2yy'
or alternatively: f(y)' = 2y dydx
Again, all we did was differentiate with respect to y and multiply by dydx
Explicit
Let's also find the derivative using the explicit form of the equation.
- To solve this explicitly, we can solve the equation for y
- Then differentiate
- Then substitute the equation for y again
Example: x2 + y2 = r2
We get the same result this way!
You can try taking the derivative of the negative term yourself.
Chain Rule Again!
Yes, we used the Chain Rule again. Like this (note different letters, but same rule):
dydx = dydfdfdx
Substitute in f = (r2 − x2):
ddx(f½) = ddf(f½)ddx(r2 − x2)
Derivatives:
ddx(f½) = ½(f−½) (−2x)
And substitute back f = (r2 − x2):
ddx(r2 − x2)½ = ½((r2 − x2)−½) (−2x)
And we simplified from there.
Using The Derivative
OK, so why find the derivative y' = −x/y ?
Well, for example, we can find the slope of a tangent line.
Example: what is the slope of a circle centered at the origin with a radius of 5 at the point (3,4)?
No problem, just substitute it into our equation:
dydx = −x/y
dydx = −3/4
And for bonus, the equation for the tangent line is:
y = −3/4 x + 25/4
Another Example
6.1 Implicit Vs Explicitap Calculus Solver
Sometimes the implicit way works where the explicit way is hard or impossible.
Example: 10x4 - 18xy2 + 10y3 = 48
How do we solve for y? We don't have to!
- First, differentiate with respect to x (use the Product Rule for the xy2 term).
- Then move all dy/dx terms to the left side.
- Solve for dy/dx
Like this:
(the middle term is explained below)
And we get:
dydx = | 9y2 − 20x3 |
3(5y2 − 6xy) |
Product Rule
For the middle term we used the Product Rule: (fg)' = f g' + f' g
6.1 Implicit Vs Explicitap Calculus Algebra
2x + 2ydydx = 0
Collect all the dydx on one side
ydydx = −x
Solve for dydx:
dydx = −xy
The Chain Rule Using dydx
Let's look more closely at how ddx(y2) becomes 2ydydx
The Chain Rule says:
dudx = dudydydx
Substitute in u = y2:
ddx(y2) = ddy(y2)dydx
And then:
ddx(y2) = 2ydydx
Basically, all we did was differentiate with respect to y and multiply by dydx
Another common notation is to use ' to mean ddx
The Chain Rule Using '
The Chain Rule can also be written using ' notation:
f(g(x))' = f'(g(x))g'(x)
g(x) is our function 'y', so:
f(y)' = f'(y)y'
f(y) = y2, so f'(y) = 2y:
f(y)' = 2yy'
or alternatively: f(y)' = 2y dydx
Again, all we did was differentiate with respect to y and multiply by dydx
Explicit
Let's also find the derivative using the explicit form of the equation.
- To solve this explicitly, we can solve the equation for y
- Then differentiate
- Then substitute the equation for y again
Example: x2 + y2 = r2
We get the same result this way!
You can try taking the derivative of the negative term yourself.
Chain Rule Again!
Yes, we used the Chain Rule again. Like this (note different letters, but same rule):
dydx = dydfdfdx
Substitute in f = (r2 − x2):
ddx(f½) = ddf(f½)ddx(r2 − x2)
Derivatives:
ddx(f½) = ½(f−½) (−2x)
And substitute back f = (r2 − x2):
ddx(r2 − x2)½ = ½((r2 − x2)−½) (−2x)
And we simplified from there.
Using The Derivative
OK, so why find the derivative y' = −x/y ?
Well, for example, we can find the slope of a tangent line.
Example: what is the slope of a circle centered at the origin with a radius of 5 at the point (3,4)?
No problem, just substitute it into our equation:
dydx = −x/y
dydx = −3/4
And for bonus, the equation for the tangent line is:
y = −3/4 x + 25/4
Another Example
6.1 Implicit Vs Explicitap Calculus Solver
Sometimes the implicit way works where the explicit way is hard or impossible.
Example: 10x4 - 18xy2 + 10y3 = 48
How do we solve for y? We don't have to!
- First, differentiate with respect to x (use the Product Rule for the xy2 term).
- Then move all dy/dx terms to the left side.
- Solve for dy/dx
Like this:
(the middle term is explained below)
And we get:
dydx = | 9y2 − 20x3 |
3(5y2 − 6xy) |
Product Rule
For the middle term we used the Product Rule: (fg)' = f g' + f' g
6.1 Implicit Vs Explicitap Calculus Algebra
Because (y2)' = 2ydydx(we worked that out in a previous example)
Oh, and dxdx = 1, in other words x' = 1
Inverse Functions
Implicit differentiation can help us solve inverse functions.
The general pattern is:
- Start with the inverse equation in explicit form. Example: y = sin−1(x)
- Rewrite it in non-inverse mode: Example: x = sin(y)
- Differentiate this function with respect to x on both sides.
- Solve for dy/dx
As a final step we can try to simplify more by substituting the original equation.
An example will help:
Example: the inverse sine function y = sin−1(x)
6.1 Implicit Vs Explicitap Calculus Calculator
We can also go one step further using the Pythagorean identity:
sin2 y + cos2 y = 1
cos y = √(1 − sin2 y )
And, because sin(y) = x (from above!), we get:
cos y = √(1 − x2)
Which leads to:
dydx= 1√(1 − x2)
Example: the derivative of square root √x
Note: this is the same answer we get using the Power Rule:
Summary
- To Implicitly derive a function (useful when a function can't easily be solved for y)
- Differentiate with respect to x
- Collect all the dy/dx on one side
- Solve for dy/dx
- To derive an inverse function, restate it without the inverse then use Implicit differentiation