Question

Find the derivative \(d x / d y\) of \(x\) with respect to \(y.\) $$x=(y-3)^{2}$$

Short answer

\(\frac{dx}{dy} = 2(y - 3)\)

Step-by-step solution

Identify the Function

The function given is in the form of a composition of functions, where the outer function is the square function and the inner function is a linear function of y. The function is given by the equation: \(x=(y-3)^{2}\).

Apply the Chain Rule

To find \(\frac{dx}{dy}\), we need to apply the chain rule, which states that if we have a composite function \( f(g(y)) \), the derivative of \(x\) with respect to \(y\) is found by taking the derivative of the outer function with respect to the inner function and multiplying it by the derivative of the inner function with respect to \(y\). So \(\frac{dx}{dy} = 2 (y - 3) \frac{d}{dy}(y-3)\).

Differentiate the Inner Function

The derivative of the inner function \((y-3)\) with respect to \(y\) is 1, because the derivative of \(y\) with respect to \(y\) is 1 and the constant \(-3\) disappears upon differentiation.

Simplify the Expression

After applying the derivatives, we have \(\frac{dx}{dy} = 2 (y - 3)(1) = 2(y - 3)\). Therefore, the derivative of \(x\) with respect to \(y\) is simply \(2(y - 3)\).

Key concepts

Derivative of Composite Functions

Understanding the derivative of composite functions is essential in calculus, particularly when dealing with more complex equations. In our exercise, we are presented with a function expressing a composition: \(x=(y-3)^{2}\). Here, \(y-3\) serves as the inner function, while squaring it represents the outer function. When taking the derivative of a composite function, we evaluate the rate of change as one function's output becomes the input for another.

The Chain Rule is our fundamental tool for handling the derivative of such composite functions. It effectively breaks down the differentiation process into manageable parts: first, differentiate the outer function, then multiply that by the derivative of the inner function. In essence, the Chain Rule tells us how to navigate through the layers of functions by differentiating from the outside in. This method ensures that all parts of the composite function are accounted for in the final derivative.

Differentiation of Polynomial Functions

Polynomial functions, which are algebraic expressions involving a sum of powers of variables multiplied by coefficients, appear frequently in calculus. The differentiation rules for polynomial functions are straightforward: for each term, multiply the coefficient by the power, then decrease the power by one. For a term like \(y^n\), its derivative with respect to \(y\) is \(ny^{n-1}\).

However, when the polynomial function is part of a composite function, we must first apply the Chain Rule before utilizing the power rule, as seen in our given exercise. It's crucial always to keep the order of operations in mind: we first apply the Chain Rule to deal with the composition, and within that process, we use the differentiation rules for polynomials on the individual components of the composite function.

Applying the Chain Rule

The Chain Rule is a critical concept in calculus for finding the derivatives of composite functions. To apply the Chain Rule effectively, one needs to identify the composite nature of a function and recognize the 'inner' and 'outer' elements within it. In the provided exercise, we recognized that \(x=(y-3)^{2}\) is a composite function with \(y-3\) as the inner function and the squaring operation as the outer function.

The application of the Chain Rule is done in two steps: First, take the derivative of the outer function with respect to the inner function—this derivative treats the inner function as if it were a variable. Then, multiply that result by the derivative of the inner function with respect to the actual variable in question, which, in our case, is \(y\). The Chain Rule enables us to elegantly handle situations where functions are nested within one another, ensuring that each part is correctly differentiated.