The derivative of a composite function can also be expressed as follows: if y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then the form... The derivative of a composite function can also be expressed as follows: if y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then the formula for the derivative is dy/dx = f'[g(x)] · g'(x).
Understand the Problem
The question is about the application of the chain rule in calculus, specifically how to derive the relationship of derivatives when dealing with composite functions. It provides a detailed mathematical approach to understanding how changes in one variable affect another through derivatives.
Answer
The derivative of a composite function is given by: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
Answer for screen readers
The relationship derived from the chain rule for the derivatives is: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
Steps to Solve
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Understand Limit Relationships We start by rewriting the limit of the derivative of $y$ with respect to $x$. The chain rule states that: $$ \lim_{\Delta x \to 0} \frac{\partial y}{\partial x} = \lim_{\Delta x \to 0} \frac{\partial y}{\partial u} \cdot \frac{\partial u}{\partial x} $$
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Use Differentiability of Functions Since $y$ is a differentiable function of $u$, and $u$ is a differentiable function of $x$, we have: $$ \lim_{\Delta x \to 0} \frac{\partial y}{\partial u} = \frac{dy}{du} \quad \text{and} \quad \lim_{\Delta x \to 0} \frac{\partial u}{\partial x} = \frac{du}{dx} $$
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Combine Results from Limits Using the results from the limits, we derive the following from the previous equations: $$ \lim_{\Delta x \to 0} \frac{\partial y}{\partial x} = \frac{dy}{du} \cdot \frac{du}{dx} $$
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Final Expression Thus, we conclude that: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
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Application to Composite Functions If we apply this to composite functions where $y = f(g(x))$, we can write: $$ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) $$
The relationship derived from the chain rule for the derivatives is: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
More Information
The chain rule is crucial in calculus for differentiating composite functions. It allows us to compute the derivative of a function based on the derivatives of its constituent functions.
Tips
- Misunderstanding Composite Functions: Students may confuse the variables and derivatives involved in composite functions. Remember that you should always identify which function is dependent on which variable.
- Ignoring Differentiability: Not recognizing that both functions need to be differentiable can lead to incorrect conclusions about limits.
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