91Ó°ÊÓ

Understanding the derivative of composite functions is crucial when dealing with more complex calculations in calculus. A composite function is created when one function is nested within another. For example, if we have two functions, g(x) and h(x), the composite function f(x) can be defined as f(x) = g(h(x)). To find the derivative of such a composite function, one cannot simply differentiate g(x) and h(x) separately.

Instead, we use the chain rule, which is a powerful differentiation technique that allows us to handle these composite functions effectively. The process involves differentiating the outer function g(x) with respect to its inner function h(x), then multiplying that result by the derivative of the inner function h(x) with respect to x. We denote this process as \( f'(x) = g'(h(x)) \times h'(x) \).

When applied to our example, with the outer function being a square root and the inner function being \( 2 + 3\tan(2x) \), we first note that the derivative of the square root function g(x) with respect to x is \( g'(x) = \frac{1}{2}x^{-\frac{1}{2}} \), assuming \( x > 0 \). We apply the chain rule to differentiate the entire composite function, leading to a precise and meticulous calculation that considers the interplay between the inner and outer functions.

The chain rule is an essential tool within calculus, especially when it comes to finding derivatives of composite functions. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function and multiplied by the derivative of the inner function.

If we denote the chain rule mathematically, it appears as \( (f(g(x)))' = f'(g(x)) \times g'(x) \). Applying this to our problem, where f(x) and g(x) are differentiable functions, we managed to find the derivative of the outer function \( g'(h(x)) \) and the derivative of the inner function \( h'(x) \) separately. These two components are then multiplied to give us the derivative of the composite function. This process simplifies what could be a complex differentiation task into manageable steps.

It's important to correctly identify the inner and outer functions and to accurately differentiate them separately before applying the chain rule. Mistakes can often occur if these steps are not followed with care. This rule is indispensable, especially when functions are layered within one another, revealing the beautiful nested structure of mathematical functions.

Trigonometric functions, such as sine, cosine, and tangent, are pervasive in calculus. Understanding their derivatives is essential for solving many calculus problems, including those involving composite functions.

For instance, one of the fundamental derivatives in trigonometry is that of the tangent function, which is \( \frac{d}{dx}(\tan(x)) = \sec^2(x) \). This becomes particularly important when the tangent function is part of a composite function, as the derivative will play a role in the application of the chain rule.

In our exercise, we encounter \( 3\tan(2x) \) as part of the inner function h(x). When differentiating with respect to x, we use the knowledge that the derivative of \( \tan(kx) \), where k is a constant, is \( k\sec^2(kx) \) due to the chain rule. Thus, for \( h'(x) = 3\frac{d}{dx}(\tan(2x)) \), we end up with \( 3(2)\sec^2(2x) \), or simply \( 6\sec^2(2x) \).

Being adept at differentiating trigonometric functions is instrumental when they are nested within other functions, as in the case of composite functions. It’s one of the fundamental skills needed to tackle a wide array of problems in calculus.