I am having trouble figuring out how to do partial derivatives. I have tried practice problems and have read my textbook but I am still so confused. Could someone please explain how to do this? Thank you.
Partial derivatives. Treat all other variables as constants.
example
f= x sin z
df/dz= x cos z and the partial df/dx= sinz
example
f= x^2y^3 z^-7
df/dy= 3x^2*y^2*z^-7
http://en.wikipedia.org/wiki/Partial_derivative
Of course! I'd be happy to help you understand partial derivatives.
To begin, let's first review what a derivative is. In calculus, a derivative measures how a function changes as its input varies. It provides us with information about the rate at which a function is changing at a particular point.
Now, a partial derivative is a specific type of derivative that deals with functions of multiple variables. It allows us to understand how a function changes with respect to one variable while keeping all other variables constant.
To calculate a partial derivative, follow these steps:
1. Identify the variable with respect to which you want to take the partial derivative. Let's denote this variable by x.
2. Treat all the other variables as constants. Consider them fixed while finding the derivative.
3. Now, differentiate the function with respect to x, just as you would with a normal derivative. Treat all other variables (held constant) as constants in the differentiation process.
4. Simplify the resulting expression as much as possible. This may involve combining like terms or applying any algebraic rules.
To illustrate this process, let's work through an example. Suppose we have the function f(x, y) = x^2 + 2xy.
To find ∂f/∂x (the partial derivative of f with respect to x) while treating y as a constant, we differentiate the function with respect to x:
∂f/∂x = 2x + 2y (since the derivative of x^2 is 2x, and the derivative of 2xy with respect to x is 2y)
Similarly, we can find ∂f/∂y (the partial derivative of f with respect to y) while treating x as a constant:
∂f/∂y = 2x + 2y (since the derivative of 2xy with respect to y is 2x)
Note that in this case, the partial derivatives with respect to x and y are equal. However, this might not always be the case, as the rates of change in different directions can vary.
Keep practicing with different functions and variables, and it will become easier to grasp the concept of partial derivatives. Remember, it's essential to understand the basic rules of differentiation before delving into partial derivatives.
If you need further assistance or more examples, feel free to ask!