The Chain Rule in Leibniz Notation Examples

Let z = y⁴ and y = 3x + 5. What is ?

We can think of z as z = (y)⁴, where y = 3x + 5. Then

Since z = y⁴,

Since y = 3x + 5,

.

Putting this information back into the chain rule,

This is almost the answer, but since we're asked for  we want to give the final answer in terms of x. We plug in 3x + 5 for y to find

That's all!

Let z = cos(sin x). Find the derivative of z with respect to x.

Think of z as cos (y) where y = sin x. We want to find  so we'll use the chain rule.

If we have

q = f(r) and r = g(i), what does the chain rule say?

The variable that keeps track of the inside function needs to be the one that "cancels out" in the chain rule:

The input to the inside function, i, will occur only in denominators:

and the outermost variable, q, will occur only in numerators:

Let u = 5z + z² and let y = ln u. Find 

u is the inside function and z is the innermost variable, so the form of the chain rule we want is

The derivative of y = ln u with respect to u is

and the derivative of u = 5z + z² with respect to z is

5 + 2z.

Applying the chain rule we find

Now substitute 5z + z² for u so that there's only one variable.

Let q = sin(6s + s^-1). Find .

This problem doesn't specify the inside function, so we'll make a name for it. Let

y = 6s + s^-1.

Then

q = sin(y).

We want to use the version of the chain rule that says

The derivative of q = sin(y) with respect to y is

cos(y)

and the derivative of y = 6s + s^-1 with respect to s is

6 – s^-2

We find this: