91Ó°ÊÓ

The chain rule is an essential concept when dealing with derivatives, especially when the function you are differentiating is composed with other functions. Simply put, the chain rule is used to differentiate a function that is nested inside another function.
For our exercise, we have the function \( H(t) = \tan 3t \). This is a composition of the outer function \( \tan(x) \) and the inner function \( 3t \).
To apply the chain rule, it's important to remember the formula: if you have a composite function \( f(g(t)) \), then the derivative is \( f'(g(t)) \cdot g'(t) \).
So, in our example, the outer derivative is \( \sec^2(3t) \) since the derivative of \( \tan(x) \) is \( \sec^2(x) \), and the inner derivative, which is the derivative of \( 3t \), is \( 3 \). Combining these, we get \( 3 \sec^2(3t) \). Using the chain rule allows us to properly handle changes to both the inner and outer functions.

Trigonometric functions, such as \( \tan(x) \), \( \sin(x) \), and \( \cos(x) \), are fundamental in calculus, especially when it comes to deriving their derivatives. Each of these functions has well-known derivatives:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \(-\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

Understanding these derivatives is crucial because they allow us to differentiate composed functions like \( \tan 3t \).
In the given problem, after applying the chain rule, the first derivative comes out as \( 3\sec^2(3t) \). Knowing that \( \tan(x) \)'s derivative is \( \sec^2(x) \) simplifies the process, as it turns a potential stumbling block into a straightforward computation.

The second derivative is the derivative of the derivative. It provides information about the change in rate of change, offering insights into the curvature and concavity of a function's graph. To find the second derivative, you follow the same differentiation rules on the first derivative. In the context of our example, we start with the first derivative: \( H'(t) = 3 \sec^2(3t) \).
Applying the chain rule again is necessary because the function \( \sec^2(3t) \) is itself a composite function. When differentiating \( \sec^2(x) \), the rule \( 2\sec^2(x)\tan(x) \) is applied.
Thus, \( H''(t) = 18 \sec^2(3t) \tan(3t) \) is derived by applying the chain rule, incorporating the derivative of \( 3t \), which is itself \( 3 \). The second derivative gives us detailed information on how \( H(t) \) is accelerating at different points, often used in motion analysis to determine concavity and potential inflection points.