# Difference between revisions of "022 Sample Final A, Problem 1"

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<span class="exam">Find all first and second partial derivatives of the following function, and demostrate that the mixed second partials are equal for the function <math> f(x, y) = \frac{2xy}{x-y}.</math> | <span class="exam">Find all first and second partial derivatives of the following function, and demostrate that the mixed second partials are equal for the function <math> f(x, y) = \frac{2xy}{x-y}.</math> | ||

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+ | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Foundations: | ||

+ | |- | ||

+ | |1)Which derivative rules do you have to use for this problem? | ||

+ | |- | ||

+ | |2)What is the partial derivative of xy, with respect to x? | ||

+ | |- | ||

+ | |1)You have to use the quotient rule, and product rule. The quotient rule says that <math>\frac{\partial}{\partial x} \left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - g'(x)f(x)}{g(x)^2}</math>, so <math>\frac{\partial}{\partial x} \left(\frac{x^2}{x + 1}\right) = \frac{2x(x + 1) - x^2}{(x + 1)^2}</math>. The product rule says <math>\frac{\partial}{\partial x} f(x)g(x) = f'(x)g(x) + g'(x)f(x) </math>. This means <math>\frac{\partial}{\partial x} x(x + 1) = (x + 1) + x </math> | ||

+ | |- | ||

+ | |2) The partial derivative is y, since we treat anything not involving x as a constant and take the derivative with respect to x. So <math>\frac{\partial}{\partial y} xy = x\frac{\partial}{\partial y} y = x.</math> | ||

+ | |} | ||

+ | |||

+ | '''Solution:''' | ||

+ | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Foundations: | ||

+ | |- | ||

+ | |The word 'marginal' should make you immediately think of a derivative. In this case, the marginal is just the partial derivative with respect to a particular variable. | ||

+ | |- | ||

+ | |The teacher has also added the additional restriction that you should not leave your answer with negative exponents. | ||

+ | |} |

## Revision as of 08:45, 5 June 2015

Find all first and second partial derivatives of the following function, and demostrate that the mixed second partials are equal for the function

Foundations: |
---|

1)Which derivative rules do you have to use for this problem? |

2)What is the partial derivative of xy, with respect to x? |

1)You have to use the quotient rule, and product rule. The quotient rule says that , so . The product rule says . This means |

2) The partial derivative is y, since we treat anything not involving x as a constant and take the derivative with respect to x. So |

**Solution:**

Foundations: |
---|

The word 'marginal' should make you immediately think of a derivative. In this case, the marginal is just the partial derivative with respect to a particular variable. |

The teacher has also added the additional restriction that you should not leave your answer with negative exponents. |