Section 3.7 and 3.8: Applications of The Chain Rule.
The Chain rule gives way to three applications which will allow for more derivative rules/tricks. They are as follows:
Application #1: Implicit Differentiation.
So far, all the functions we have done are explicit. A function y = f (x) is defined for a specific variable x.
Sometimes, however, it is nearly impossible to write a function as a function of a variable. Here’s an example:
x3 y 2 + 3xy = cos(xy). If you want to write this as y = f (x), it would be a hard algebraic task.
The way to treat an equation like this is by treating y as a function of x, and deriving each side of the equation
accordingly. Here’s an example: x2 + y 2 = 20.
To compute this derivative, you take the derivative of each side of the equation. The derivative of x2 with respect
to x is 2x. The derivative of 20 with respect to x is 0. And finally, the derivative of y 2 with respect to x is
dy
dy
2y · dx
. You then solve for dx
.
Exercises: Compute dy/dx for the following equations.
(a) x3 + y 3 = 18xy
(b) (3y + 7x)2 = 6y
2
(d) x + sin(y) = 8y
(e) 2ex = 2x + 3y 3 − 1
(c) x4 +
√
y = cot(y)
This of course leads to tangent lines as well.
Example: Compute the tangent line for the following equation at the given point:
2
2ex = 2x + 3y 3 − 1 at the point (0, 1).
Example: Compute the second derivative,
d2 f
for the function given below:
dx2
x + y 3 = 5e3y
Application #2: Logarithmic Differentiation
The third application of the Chain Rule involves simplification of your equation using logarithmic properties. We
start with a recap:
Properties of Logarithms: Given that x, y, b > 0 and that b 6= 1, we have:
• logb (xy) = logb (x) + logb (y)
• logb (x/y) = logb (x) − logb (y)
• logb (xn ) = n logb (x)
• If logb (x) = y, then by = x.
Note that these properties go forwards and backwards. These will be useful when doing logarithmic differentiation. Note also that this applies if we change the base to base e, meaning these properties work for ln.
We can use the Chain Rule to compute the derivative of f (x) = ln(x):
f (x) = ln(x)
ef (x) = eln(x)
Exponentiate both sides.
ef (x) = x
Cancellation Property on the right side.
ef (x) · f 0 (x) = 1
Take the derivative using Chain Rule.
eln(x) · f 0 (x) = 1
Substitute f (x) = ln(x).
x · f 0 (x) = 1
Cancellation property on left side.
1
0
f (x) =
This gives us the derivative of ln(x).
x
Thus, we now have a rule for taking derivatives of ln terms.
Examples: Compute the derivative of the following logarithmic functions:
f (x) = ln(5x2 − 3x + 1)
g(x) = 7 ln(9x + sin(x))
Logarithmic differentiation is a shortcut method that combines many concepts to make differentiation a more
fluid and efficient process. The first step in log differentiation is simplification of the function or equation. You
take ln of both sides and use the properties of logs to clean up the equation. Then use implicit differentiation.
IMPORTANT: Often times, other rules may be used to bypass logarithmic differentiation. However, if a variable
is in the exponent of your equation or function, then this is the only way (that we’ve learned so far) to take the
derivative.
Examples: Solve for dy/dx for the following equations:
√
(a) 4y 5 = (x − 5)3 5x2 + 1
3
(c) y = 5xx + 8x
(b) 5x2 y 3 =
x4
5x2 − 1
(d) 5y = (sin(x))9x
Miscellaneous Exponential and Logarithmic Functions:
Remember that exponential functions do not need to have e as their base. It can be any positive number. The
same can be said about logarithmic functions. Let us now explore f (x) = ax and g(x) = loga (x), where a is any
positive base.
Examples: Compute the derivative of the following functions:
(a) f (x) = 7x
(b) g(x) = log6 (x)
To do part (a), notice that you have a variable in your exponent, so you will want to use log differentiation. To
ln(x)
.
do part (b), you will need to use another logarithmic property: log6 (x) =
ln(6)