f(x,y)=x^2 e^2x lny (2,1)
I need to find the partial derivatives with respect to x and y. I have not idea how to do this especially with the (2,1).
f(x,y)=(x^2)(e^2x)lny
f’x=((x^2)(e^2x))’lny= ((x^2)’(e^2x)+x^2(e^2x)’)lny=(2xe^2x+2x^2e^2x)lny=2xe^2x(1+x)lny
f’y=(x^2e^2x)(lny)’=x^2e^2x/y
To find the partial derivatives of f(x,y) with respect to x and y, you can use the chain rule.
1. Partial derivative with respect to x (assuming y is constant):
To find ∂f/∂x, we treat y as a constant and differentiate f(x,y) with respect to x. To do this, we apply the product rule and chain rule.
First, let's calculate the partial derivative of the function f(x,y) with respect to x, assuming y is a constant:
∂f/∂x = ∂(x^2 e^(2x) lny)/∂x
Using the product rule, we can break down the function as follows:
∂f/∂x = ∂(x^2)/∂x * e^(2x) lny + x^2 * ∂(e^(2x))/∂x * lny + x^2 e^(2x) * ∂(lny)/∂x
Simplifying, we have:
∂f/∂x = 2x * e^(2x) lny + x^2 * (2e^(2x)) * lny + x^2 e^(2x) * ∂(lny)/∂x
To calculate the last term, we need to know the partial derivative of lny with respect to x. Since y is a constant with respect to x, the derivative of lny with respect to x is 0.
Therefore, the partial derivative of f(x,y) with respect to x is:
∂f/∂x = 2x * e^(2x) lny + 2x^2 e^(2x) lny
2. Partial derivative with respect to y (assuming x is constant):
To find ∂f/∂y, we treat x as a constant and differentiate f(x,y) with respect to y. Since x is constant, the terms involving x will disappear.
∂f/∂y = ∂(x^2 e^(2x) lny)/∂y
Differentiating with respect to y, we get:
∂f/∂y = x^2 e^(2x) * ∂(lny)/∂y
Since y is a function of y, we can calculate the partial derivative of lny with respect to y:
∂(lny)/∂y = 1/y
Therefore, the partial derivative of f(x,y) with respect to y is:
∂f/∂y = x^2 e^(2x) * (1/y)
Now, to evaluate the partial derivatives at the point (2,1), substitute the values of x and y into the resulting expressions:
∂f/∂x = 2(2) * e^(2(2)) * ln(1) + 2(2)^2 * e^(2(2)) * ln(1) = 8e^4
∂f/∂y = (2^2) * e^(2(2)) * (1/1) = 4e^4
Hence, at the point (2,1), the partial derivative with respect to x is 8e^4 and the partial derivative with respect to y is 4e^4.
To find the partial derivatives of the function f(x,y) = x^2 * e^(2x) * ln(y) at the point (2,1), we will calculate the derivative of the function with respect to each variable separately.
Partial derivative with respect to x:
To find the partial derivative of f(x,y) with respect to x, treat y as a constant and differentiate the expression with respect to x. The rule for differentiating a function with multiple terms involves finding the derivative of each term separately and then adding them.
For the first term, x^2, the derivative is found using the power rule: d/dx (x^n) = nx^(n-1). So, the derivative of x^2 with respect to x is 2x.
For the second term, e^(2x), we apply the chain rule, which states that if we have a composition of functions, we multiply the derivative of the outer function with the derivative of the inner function. The derivative of e^u where u = 2x is d/du (e^u) = e^u. Since u = 2x, we use the chain rule and multiply this result by the derivative of the inner function, which is 2. Thus, the derivative of e^(2x) with respect to x is 2e^(2x).
For the third term, ln(y), we differentiate using the rule d/dx (ln(u)) = 1/u * du/dx. In this case, u = y and du/dx = 0 (since we're treating y as a constant). Thus, the derivative of ln(y) with respect to x is 0.
Now, we can calculate the partial derivative of f(x,y) with respect to x by multiplying the derivatives of each term:
∂f/∂x = (2x) * (e^(2x)) * ln(y) + x^2 * (2e^(2x)) * ln(y) * 0
= 2x * e^(2x) * ln(y)
Partial derivative with respect to y:
To find the partial derivative of f(x,y) with respect to y, treat x as a constant and differentiate the expression with respect to y.
For the first term, x^2, the derivative is 0 since x is treated as a constant.
For the second term, e^(2x), the derivative is also 0 since x is treated as a constant.
For the third term, ln(y), we can differentiate using the rule d/dy (ln(u)) = 1/u * du/dy. Here, u = y, and the derivative of y with respect to y is 1. Thus, the derivative of ln(y) with respect to y is 1/y.
Now, we can calculate the partial derivative of f(x,y) with respect to y:
∂f/∂y = x^2 * e^(2x) * (1/y)
= (x^2 * e^(2x))/y
Finally, to evaluate these partial derivatives at the specified point (2,1), substitute x = 2 and y = 1 into the expressions we obtained:
∂f/∂x = 2(2) * e^(2(2)) * ln(1) = 4e^4 * ln(1) = 4e^4 * 0 = 0
∂f/∂y = (2^2 * e^(2*2))/1 = 4e^4
Therefore, the partial derivative with respect to x is 0, and the partial derivative with respect to y is 4e^4.