Evaluate: f(x,y)=2x^3e^y
a) partial derivative with respect to x.
I know that you have to treat y as a constant but I have no idea what to do. I do not understand partial derivatives at all. Please help!!
b) partial derivative with repsect to y.
df/dx = 2 e^y (3 x^2)
= 6 x^2 e^y
df/dy = 2 x^3 (de^y/dy) = 2 x^3 e^y (in other words partial with respect to y is original function)
for the first one how did you get (3x^2)?
To evaluate the partial derivative of a function with respect to a specific variable, you can follow these steps:
a) To find the partial derivative of f(x,y) with respect to x, you need to treat y as a constant while differentiating the function with respect to x.
1. Begin by differentiating each term of the function separately with respect to x. In this case, the function has two terms: 2x^3 and e^y.
Differentiating the first term, 2x^3, with respect to x:
- The power rule states that if you have a term in the form of x^n, where n is a constant, the derivative is given by n times x^(n-1).
- Therefore, the derivative of 2x^3 with respect to x is 2 * 3 * x^(3-1) = 6x^2.
Differentiating the second term, e^y, with respect to x:
- Since y is treated as a constant, differentiating e^y with respect to x results in zero.
2. Finally, combine the derivatives of each term to get the partial derivative of the function with respect to x:
- 6x^2 + 0 = 6x^2
So, the partial derivative of f(x,y) with respect to x is 6x^2.
b) To find the partial derivative of f(x,y) with respect to y, you treat x as a constant while differentiating the function with respect to y.
1. Differentiate each term of the function separately with respect to y, treating x as a constant:
- The derivative of 2x^3 with respect to y is zero since x is treated as a constant.
- The derivative of e^y with respect to y is e^y.
2. Combine the derivatives of each term to get the partial derivative of the function with respect to y:
- 0 + e^y = e^y
So, the partial derivative of f(x,y) with respect to y is e^y.