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The Chain Rule in Leibniz Notation Examples

Example 1

Let z = y4 and y = 3x + 5. What is ?

Example 2

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

Example 3

If we have

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

Example 4

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

Example 5

Let q = sin(6s + s-1). Find .