Evaluate the first partial derivatives of the function at the given point.(0,e)
f(x,y) = 6e^x ln(y)
fx(0,e) = ?
fy(0,e) = ?
To evaluate the first partial derivatives of the function at the given point (0, e), we need to calculate the derivative with respect to x (fx) and the derivative with respect to y (fy) separately.
To find fx(0, e), we differentiate f(x, y) with respect to x while treating y as a constant.
Step 1: Calculate the derivative of 6e^x ln(y) with respect to x.
f'(x, y) = d/dx[6e^x ln(y)]
= 6 * d/dx[e^x ln(y)]
= 6 * e^x * ln(y) * d/dx[e^x] (Using the product rule)
= 6 * e^x * ln(y) * e^x (Differentiating e^x)
Step 2: Substitute x = 0 and y = e into the derived expression.
fx(0, e) = 6 * e^0 * ln(e) * e^0
= 6 * 1 * 1 * 1
= 6
Therefore, fx(0, e) = 6.
To find fy(0, e), we differentiate f(x, y) with respect to y while treating x as a constant.
Step 1: Calculate the derivative of 6e^x ln(y) with respect to y.
f'(x, y) = d/dy[6e^x ln(y)]
= 6e^x * d/dy[ln(y)]
= 6e^x * 1/y (Using the derivative of ln(y))
Step 2: Substitute x = 0 and y = e into the derived expression.
fy(0, e) = 6e^0 * 1/e
= 6 * 1 * 1/e
= 6/e
Therefore, fy(0, e) = 6/e.