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# The Chain Rule in Leibniz Notation Exercises

### Example 1

• If z = ln(x5 + 4x2), then what is the derivative of z with respect to x?

### Example 2

• If z = (sin x + cos x)7 what is the derivative of z with respect to x?

### Example 3

• What is the derivative of z with respect to x if ?

### Example 4

•  If z = e12x + 4 what is the derivative of z with respect to x?

### Example 5

• If , what is the derivative of z with respect to x?

### Example 6

For the pair of functions, determine what the chain rule says.

• Let p = f(s) and s = g(z).

### Example 7

For the pair of functions, determine what the chain rule says.

• Let r = g(t) and q = f(r).

### Example 8

For the pair of functions, determine what the chain rule says.

• Let x = f(y) and y = g(z).

### Example 9

• If u = es and s = 5 – 3x2, find .

### Example 10

• If s = t4 and t = r4 + 1, find .

### Example 11

• If m = sin n and z = ln m, find .

### Example 12

• If z = (x4 + 4x) and y = z3 + z5, find .

### Example 13

• If  and q = tan p, find .

### Example 14

• What is given that r = sin(x2 + 1) + cos(x2 + 1)?

### Example 15

• What is if p = er2 + 3r?

### Example 16

• Find  given that y = ln(z4 + z3).

### Example 17

• Find  given that .

### Example 18

• Find  given that q = 1412sin t.