Partial derivative examples

# Multivariable calculus and vector analysis

## A set of on-line readings

### Non-Math ML version

### Partial derivative examples

#### Example 1

Let f(x,y) = y3x2. Calculate (x,y).

Solution: To calculate (x,y), we simply view y as being a fixed number and calculate the ordinary derivative with respect to x. The first time you do this, it might be easiest to set y = b, where b is a constant, to remind you that you should treat y as though it were number rather than a variable. Then, the partial derivative (x,y) is the same as the ordinary derivative of the function g(x) = b3x2. Using the rules for ordinary differentiation, we know that

(x) = 2b3x. |  |

Now, we remember that b = y and substitute y back in to conclude that

(x,y) = 2y3x. |  |

#### Example 2

For the same f, calculate (x,y).

Solution: This time, we’ll just calculate the derivative with respect to y directly without replacing x with a constant. We just have to remember to treat x like a constant and use the rules for ordinary differentiation. We don’t touch the x2 and only differentiate the y3 factor to calculate that

(x,y) = 3x2y2. |  |

#### Example 3

For the same f, calculate (1, 2).

Solution: From example 1, we know that (x,y) = 2y3x. To evaluate this partial derivative at the point (x,y) = (1, 2), we just substitute the respective values for x and y:

(1, 2) = 2(23)(1) = 16. |  |

#### Example 4

For

f(x1,x2,x3,x4) = 3 + 5x1x3x4 |  |

calculate (a,b,c,d).

Solution: Although this initially looks hard, it’s really any easy problem. The ugly term does not depend on x3, so in calculating partial derivative with respect to x3, we treat it as a constant. The derivative of a constant is zero, so that term drops out. The derivative is just the derivative of the last term with respect to x3, which is

(x1,x2,x3,x4) = 5x1x4 |  |

Substituting in the values (x1,x2,x3,x4) = (a,b,c,d), we obtain the final answer

(a,b,c,d) = 5ad. |  |

## Contents by lecture

### Part 1

### Part 2

### Part 3

### Part 4

### Part 5

- Introduction to partial derivatives*
- Quiz 5: Partial derivatives intro quiz*
- Partial derivatives by limit definition
- Partial derivative examples

### Part 6

### Part 7

### Part 8

### Part 9

### Part 10

### Part 11

### Part 12

### Part 13

### Part 14

### Part 15

### Part 16

### Part 17

### Part 18

### Part 19

### Part 20

### Part 21

### Part 22

### Part 23

### Part 24

### Part 25

### Part 26

### Part 27