Problem 9 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the following functions. $$f(x, y)=3 x^{2} y+2$$

Step-by-Step Solution

Answer

Question: Find the first partial derivatives of the function $$f(x, y) = 3x^2y + 2$$. Answer: The first partial derivatives are $$\frac{\partial f}{\partial x} = 6xy$$ and $$\frac{\partial f}{\partial y} = 3x^2$$.

Step 1: Differentiate with respect to x

Keep y constant and find the derivative of the function with respect to x: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(3x^2y + 2)$$

Step 2: Compute the x partial derivative

Applying the power and constant rule for derivatives, we have: $$\frac{\partial f}{\partial x} = 6xy$$

Step 3: Differentiate with respect to y

Keep x constant and find the derivative of the function with respect to y: $$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(3x^2y + 2)$$

Step 4: Compute the y partial derivative

Applying the power and constant rule for derivatives, we have: $$\frac{\partial f}{\partial y} = 3x^2$$

Step 5: Write down the first partial derivatives

The first partial derivatives of the function $$f(x, y) = 3x^2y + 2$$ are: $$\frac{\partial f}{\partial x} = 6xy$$ $$\frac{\partial f}{\partial y} = 3x^2$$