Question

Derivative of a composite function. For \(f(x, y)=\) \(x^{2} y+y^{3}\), where \(y=m r^{2}\), find \(d f / d r\).

Short answer

\[ \frac{d f}{d r} = (x^2 + 3 m^2 r^4) \cdot 2 m r \]

Step-by-step solution

- Write down the given functions

We have a composite function where the outer function is given by: \[ f(x, y) = x^2 y + y^3 \] and the inner function is given by: \[ y = m r^2 \]

- Apply the chain rule

To find \( \frac{d f}{d r} \), we use the chain rule. This means we need to calculate partial derivatives: \[ \frac{d f}{d r} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial r} \]

- Calculate \( \frac{\partial f}{\partial x} \)

Differentiate \( f(x, y) \) with respect to \( x \): \[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 y + y^3) = 2x y \]

- Calculate \( \frac{\partial x}{\partial r} \)

Since \( x \) is not a function of \( r \), \( \frac{\partial x}{\partial r} = 0\).

- Calculate \( \frac{\partial f}{\partial y} \)

Differentiate \( f(x, y) \) with respect to \( y \): \[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2 y + y^3) = x^2 + 3y^2 \]

- Calculate \( \frac{\partial y}{\partial r} \)

Differentiate \( y = m r^2 \) with respect to \( r \): \[ \frac{\partial y}{\partial r} = 2 m r \]

- Combine the results

Substitute the obtained partial derivatives into the chain rule formula: \[ \frac{d f}{d r} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial r} = 2x y \cdot 0 + (x^2 + 3 y^2) \cdot 2 m r \]

- Simplify the expression

Since \( 2x y \cdot 0 = 0 \), we simplify the expression to: \[ \frac{d f}{d r} = (x^2 + 3 y^2) \cdot 2 m r \]

- Substitute \( y = m r^2 \) into the expression

Simplify further to obtain: \[ \frac{d f}{d r} = (x^2 + 3 m^2 r^4) \cdot 2 m r \]

Key concepts

chain rule in calculus

Understanding the chain rule is crucial for finding the derivative of composite functions. The chain rule allows us to differentiate two or more functions nested within each other.
The general form of the chain rule is: \[\frac{d}{dx}f(g(x)) = f'(g(x)) \times g'(x)\]
For multivariable functions, the chain rule involves partial derivatives:
\[ \frac{d f}{d r} = \frac{\bold{abla} f}{abla x} \frac{abla x}{abla r} + \frac{abla f}{abla y} \frac{abla y}{abla r} \]
It breaks down the derivative into simpler parts and then combines them. This is highly useful when dealing with functions dependent on multiple variables.

partial derivatives

Partial derivatives focus on the change of a function in terms of one variable, keeping other variables constant. If we have a function of two variables, like \[f(x,y)\], we write the partial derivatives with respect to \[x\] and \[y\] as \[ \frac{abla f}{abla x}\] and \[\frac{abla f}{abla y} \].
Here's how you compute them:

• To find \[\frac{abla f}{abla x}\], take the derivative of the function treating \[y\] as a constant.
• To find \[\frac{abla f}{abla y}\], take the derivative of the function treating \[x\] as a constant.

This breakdown simplifies the differentiation process, allowing us to handle more complex functions.
For example, for \[f(x, y) = x^2 y + y^3\]:
When differentiating with respect to \[x\]:
\[ \frac{abla f}{abla x} = 2x y \]
When differentiating with respect to \[y\]:
\[ \frac{abla f}{abla y} = x^2 + 3 y^2 \]

composite function

Composite functions involve two or more functions combined such that the output of one function becomes the input of another. For our exercise, we have:

• An outer function: \[f(x, y) = x^2 y + y^3\].
• An inner function: \[y = m r^2\].

Applying these in solving \[ \frac{d f}{d r} \], we first recognize that the function \[f(x, y)\] depends on \[y\], which itself is a function of \[r\].
This nesting of functions is why the chain rule becomes essential, breaking down the process as:

• Differentiate \[f\] with respect to \[x\] and \[y\].
• Differentiate \[y\] with respect to \[r\].
• Combine the results for the final derivative \[ \frac{d f}{d r} \].

This structured approach simplifies and aids in understanding how the derivative of more complex, nested functions is determined.