Transcript Implicit Differentiation - Dr. Phong Chau's home page

Section 3.5
Implicit Differentiation
1
Example
If f(x) = (x7 + 3x5 – 2x2)10, determine f ’(x).
Answer: f΄(x) =10(x7 + 3x5 – 2x2)9(7x6 + 15x4 – 4x)
Now write the answer above only in terms of y
if y = x7 + 3x5 – 2x2.
Answer: f ΄(x) = 10y9y΄
Examples
If y is some unknown function of x, find
d 10
y 

dx
d
y
3e 

dx
d 2 3
x y 

dx
Purpose
9x +
x2
– 2y = 5
Easy to solve for y
and differentiate
5x – 3xy + y2 = 2y
Not easy to solve for y
and differentiate
In equations like 5x – 3xy + y2 = 2y, we simply assume that y =
f(x), or some function of x which is not easy to find.
Process wise, simply take the derivative of each side of the
equation with respect to x and when we encounter terms
containing y, we use the chain rule.
Example
y3 = 2x
3y2  y '  2
Solving for y’, we have the derivative
2
y'  2
3y
Example
x2y3 = -7
2 xy3  x2  3 y 2  y '  0
Solving for y’, we have
3x2 y 2  y '  2 xy3
2 xy3 2 y
y'  2 2 
3x y
3x
Implicit Differentiation
•
Differentiate both sides of the equation.
Since y is a function of x, every time we differentiate a term
containing y, we need to multiply it by y’ or dy/dx.
•
Solve for y’.
•
•
•
Every term containing y’ should be moved to the left by
adding or subtracting terms only.
Every term containing no y’ should be moved to the right
hand side.
Factor out y’ and divide both sides by the expression
inside ( ).
Examples
Determine dy/dx for the following.
1. 3 x  2 3 y  5 y2
2. 3x2  2x2 y 2  5 y
3. sin  y 3   x 2  2 y
Find the equation of tangent line to the curve.
x2  y 2  100 ;
(8, 6)
y  2x y  8 y  x  19 ; x  2
3
2
3
Example
Find the derivative for
y  arccos x
cos y  x
 sin y  y  1
1
y 
sin y
y 
1
1  x2
Derivative of Trig functions
d
1
[arcsin x] 
dx
1 x2
d
1
[arccosx] 
dx
1 x2
d
1
[arctan x] 
dx
1 x2
d
1
[arc cot x] 
dx
1 x2
d
1
[arcsec x] 
2
dx
| x | x 1
d
1
[arccsc x] 
2
dx
| x | x 1
Examples
Find the derivative for each function.
y  5arctan x  4arccot x
y  arccos(tan 3x)
y  arcsec( x )
2
Examples
Find and simplify dy/dx for each function.
y  8arcsin( x )  8arccos( x )
y  arcsin x  1  x
2
y  x  arccos x  1  x
y  x arctan(5x)
2
x
y  arc cot x 
1  x2
2