Calories consumed and calories burned have an impact on our weight. Let's say that our weight, u, depended on the calories from food eaten, x, and the amount of physical exertion we do, y. If we only regulated our eating, while doing the same exercise every day, we could ask how does u change when we vary only x. Likewise, we could keep x constant and take note of how u varies when we change y. This would be like keeping a constant daily diet while changing how much we exercise. This idea of change with respect to one variable while keeping other variables constant is at the heart of the partial derivative.
When we write u = u(x,y), we are saying that we have a function, u, which depends on two independent variables: x and y. We can consider the change in u with respect to either of these two independent variables by using the partial derivative. The partial derivative of u with respect to x is written as:
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What this means is to take the usual derivative, but only x will be the variable. All other variables will be treated as constants. We can also determine how u changes with y when x is held constant. This is the partial of u with respect to y. It's written as:
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For example, if u = y * x^2, then:
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Likewise, we can differentiate with respect to y and treat x as a constant with the equation:
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The rule for partial derivatives is that we differentiate with respect to one variable while keeping all the other variables constant. As another example, find the partial derivatives of u with respect to x and with respect to y for:
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To do this example, we will need the derivative of an exponential with the following:
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And the derivative of a cosine, which is written as:
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Thus:
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and
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So far we have defined and given examples for first-order partial derivatives. Second-order partial derivatives are simply the partial derivative of a first-order partial derivative. We can have four second-order partial derivatives, which you can see right here:
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Continuing with our first example of u = y * x^2,
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and
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Likewise,
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and
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And in our second example:
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and
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The mixed partial derivatives become:
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and
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Note that we will always get:
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This can be used to check our work.
What if the variables x and y also depend on other variables? For example, we could have x = x(s,t) and y = y(s,t).
Then, as you can see, we get the partial derivatives through addition of the different partial derivatives of x and y.
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Let's work this out given the following functions:
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Find:
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We first calculate the required first-order partial derivatives:
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Then:
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Partial derivatives can be expressed using a subscript. In the case of first-order partial derivatives:
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For second-order partial derivatives:
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If you look carefully at each step in the following example, you will see why the order of the subscripts for mixed partial derivatives is reversed, which is reflected here:
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Let's very briefly review what we've learned about partial derivatives. First, we saw that partial derivatives are evaluated by treating one variable as the independent variable while keeping all other variables constant. We can take first-order partial derivatives by following the rules of ordinary differentiation.
Then we looked at how second-order partial derivatives are partial derivatives of first-order partial derivatives. We next saw how to evaluate partial derivatives when the variables are dependent on other variables. Finally, we looked at a subscript notation for expressing the partial derivative, both with the first and second orders.
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1. The function f(x, y) gives us the profit (in dollars) of a certain commodity as the number of commodities x sold and the number of days y the commodity is on the market. The function is below:
f(x, y) = (x^2) e^(-y).
Find the rate of change of the profit with respect to the number of commodities sold and the number of days the commodity stays on the market.'
2. The following function below is a product of a logarithmic function and a trigonometric function. Find its first-order, partial derivatives:
g(x, y) = ln(x) cos(4y).
1. To find the rate of change of the profit, f, with respect to the number commodities sold, x, we take the partial derivative of f with respect to x while keeping y as constant. This yields the following:
f_x = e^(-y) (x^2)' = e^(-y) 2x = 2x e^(-y) dollars per number of commodities sold.
Similarly, to find the rate of change of the profit, f, with respect to the number of days, y, that the commodity stays on the market, we calculate the partial derivative of f with respect to y while keeping x as constant. This yields the following:
f_y = [e^(-y)]' (x^2) = -e^(-y) (x^2) = -(x^2) e^(-y) dollars per day.
2. To get the first-order, partial derivative of g(x, y) with respect to x, we differentiate g with respect to x, while keeping y constant. This leads to the following, first-order, partial derivative:
g_x = [ln(x)]' cos(4y) = (1/x) cos(4y).
Similarly, to get the first-order, partial derivative of g(x, y) with respect to y, we differentiate g with respect to y, while keeping x constant. This leads to the following, first-order, partial derivative:
g_y = ln(x) [cos(4y)]' = ln(x) -4 sin(4y) = -4 ln(x) sin(4y).
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