Finding the Partial Derivative of z x^2y - x sin(xy)

When dealing with partial derivatives, we often encounter functions that require a careful application of both the product rule and the chain rule. In this article, we will walk through how to find the partial derivative (frac{partial z}{partial x}) for the function z x^2y - x sin(xy). This is a crucial skill for understanding and solving problems in differential calculus, which is fundamental in various fields such as physics, engineering, and economics.

Understanding the Problem

The problem at hand is to find the partial derivative (frac{partial z}{partial x}) for the function:

(z x^2y - x sin(xy))

To solve this, we need to differentiate the function with respect to (x), treating (y) as a constant. This process will involve breaking down the function into its components and applying the necessary rules of differentiation.

Step 1: Differentiate the First Term (x^2y)

The first term is (x^2y).

(frac{partial}{partial x}(x^2y) 2xy)

Here, we used the power rule of differentiation, which states that the derivative of (x^n) is (nx^{n-1}), along with the constant multiple rule that treats (y) as a constant.

Step 2: Differentiate the Second Term (-x sin(xy))

The second term is (-x sin(xy)). To differentiate this, we apply the product rule, which states:

(frac{partial}{partial x}(uv) u'v uv')

Here, let (u -x) and (v sin(xy)). We need to find (u') and (v').

(frac{partial}{partial x}(-x) -1)

For (v'), we apply the chain rule:

(frac{partial}{partial x}sin(xy) cos(xy) cdot frac{partial}{partial x}(xy) cos(xy) cdot y)

Thus, the derivative of the second term is:

(frac{partial}{partial x}(-x sin(xy)) -1 cdot sin(xy) (-x) cdot cos(xy) cdot y -sin(xy) - xy cos(xy))

Combining the Results

Now, we combine the derivatives of the two terms:

(frac{partial z}{partial x} 2xy - sin(xy) - xy cos(xy))

This is the final result for the partial derivative of (z) with respect to (x).

Summary

The partial derivative (frac{partial z}{partial x}) of the function (z x^2y - x sin(xy)) is:

(frac{partial z}{partial x} 2xy - sin(xy) - xy cos(xy))

This result highlights the importance of applying the correct rules of differentiation, such as the product rule and the chain rule, when dealing with more complex functions. Understanding these rules is key to solving a wide range of problems in differential calculus.