91Ó°ÊÓ

When you encounter a function with multiple variables, such as f(x, y), and you're asked to find the partial derivative with respect to x, you focus on differentiating only with respect to x while treating all other variables as constants. It's akin to asking the question: How does the function's value change as x alone changes?

The function in our exercise, f(x, y) = x^2 * y^3 - 3x, includes both x and y. To find the partial derivative with respect to x, we look at each term containing x and apply the power rule for differentiation. This rule tells us that the derivative of x^n with respect to x is n*x^(n-1). For the term x^2 * y^3, treating y^3 as a constant, the derivative of x^2 is 2x. Therefore, the partial derivative of this term is 2x * y^3. The term -3x is simpler since its derivative is just -3, because the power of x is 1 and following the power rule gives us 1*x^(1-1) = 1. Combining these derivatives gives us the complete partial derivative with respect to x: \( \frac{\partial f}{\partial x} = 2x*y^3 - 3 \).

Conversely, when we want the partial derivative with respect to y, we differentiate with respect to y and treat other variables like x as constants. This helps us understand how the function changes as y changes, independent of x.

In the function f(x, y) = x^2 * y^3 - 3x, the first term x^2 * y^3 becomes the focus since it contains y. Using the power rule, we differentiate y^3 with respect to y to get 3y^2. As x^2 is treated as a constant multiplier, the partial derivative of the term is 3x^2 * y^2. The second term, -3x, does not contain y, so its partial derivative is zero. Thus, we arrive at the partial derivative with respect to y as \( \frac{\partial f}{\partial y} = 3x^2*y^2 \).

The power rule is a fundamental technique in calculus used to find derivatives of functions that contain a variable raised to a power. The rule states that if you have a function of the form x^n, where n is any real number, the derivative of that function with respect to x is n*x^(n-1). This makes it particularly simple to differentiate polynomials.

Applying this rule to the function from our exercise, f(x, y), where we have terms like x^2 and y^3, we can find the derivatives easily. For instance, the derivative of x^2 with respect to x is 2x as per the rule, which simplifies the process significantly. It's important to remember that this rule only applies to powers of variables, not constants. For a constant c, the derivative is zero, which is why the partial derivatives of constants or terms not containing the variable of differentiation result in zero in the partial differentiation process.