# The Chain Rule in Leibniz Notation Exercises

### Example 1

- If
*z*= ln(*x*^{5}+ 4*x*^{2}), 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

*z*=*e*^{12x + 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*=*e*^{s}and*s*= 5 – 3*x*^{2}, find .

### Example 10

- If
*s*=*t*^{4}and*t*=*r*^{4}+ 1, find .

### Example 11

- If
*m*= sin*n*and*z*= ln*m*, find .

### Example 12

- If
*z*= (*x*^{4}+ 4*x*) and*y*=*z*^{3}+*z*^{5}, find .

### Example 13

- If and
*q*= tan*p*, find .

### Example 14

- What is given that
*r*= sin(*x*^{2}+ 1) + cos(*x*^{2}+ 1)?

### Example 15

- What is if
*p*=*e*^{r2 + 3r}?

### Example 16

- Find given that
*y*= ln(*z*^{4}+*z*^{3}).

### Example 17

- Find given that .

### Example 18

- Find given that
*q*= 14^{12sin t}.

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