91Ó°ÊÓ

In calculus, function composition is a way of combining two functions where the output of one function becomes the input of another. In this problem, we have the expression \( x - \tan 3x \), which is a composite function. The inner function, \( g(x) \), is nested within another outer function, \( f(u) \).

By identifying the inner function, we can break down complex expressions into simpler parts, allowing us to apply calculus rules more easily. Here, we identify the innermost part of the function, \( 3x \), as \( u = g(x) \). That simplifies our expression such that \( y = f(u) = x - \tan u \). Separating functions in this way clarifies how each part interacts and makes computing derivatives more straightforward.

Key points to remember:

• Function composition involves two functions: an **outer function** and an **inner function**.
• The **inner function** \( g(x) \) is inside another function, here, \( u = 3x \).
• The result of one function (\( g(x) \)) is used as an input for the next function (\( f(u) \)).

Computing derivatives, particularly of composite functions, is an essential skill in calculus. Derivatives represent the rate at which a function changes at any given point.

In this exercise, once we've defined \( y = f(u) = x - \tan u \), we need to find \( \frac{dy}{du} \), the derivative of the outer function \( f(u) \) with respect to \( u \). This involves understanding how the tangent function changes. - The derivative of \( \tan u \) with respect to \( u \) is \( \sec^2 u \). Therefore, the derivative of our function \( f(u) \) is:
\[ f'(u) = \frac{d}{du}(x - \tan u) = -\sec^2 u \]

Once we have the derivatives of both the outer and inner functions, we can use the chain rule to find the overall derivative \( \frac{dy}{dx} \). It's all about taking it step-by-step.

In summary:

• First, identify the parts of your function with respect to which you will differentiate.
• Compute the derivative of \( f(u) \) with respect to \( u \) to find \( \frac{dy}{du} \).
• Understanding derivatives is all about recognizing how small changes in one variable affect overall function values.

Differentiating the inner function is crucial when using the chain rule in calculus. The inner function is a simple linear function here, \( g(x) = 3x \). Differentiating it with respect to \( x \) yields \( g'(x) = 3 \).

While seemingly simple, correctly identifying and differentiating the inner function ensures the correct application of the chain rule. It's like setting the stage for performing the actual chain rule, influencing the overall derivative. If this step is missed or done incorrectly, it can lead to wrong results.

To compute the composite derivative \( \frac{dy}{dx} \):

• Differentiate \( g(x) \) to find \( \frac{du}{dx} \), here resulting in \( g'(x) = 3 \).
• Multiply the derivative of the inner function \( \frac{du}{dx} \) with the derivative of the outer function \( \frac{dy}{du} \).
• Finally, express the composite function's derivative by substituting back the inner function's expression, \( u = 3x \), into the result.

This approach gives us the final derivative: \( \frac{dy}{dx} = -3 \sec^2(3x) \). As you can see, each piece of the puzzle comes together to show how everything is interconnected in calculus.