91Ó°ÊÓ

The chain rule is a powerful tool for differentiating composite functions. When you have a function composed of two or more other functions, the chain rule allows us to compute the derivative by differentiating each function separately. In the given problem, we employed the chain rule to differentiate the composite function \((1 + \cos(2t))^{-4}\).

The chain rule tells us to take the derivative of the outer function and multiply it by the derivative of the inner function. Mathematically, if you have a function \(y = f(g(t))\), the chain rule formula is:

• \(\frac{dy}{dt} = f'(g(t)) \cdot g'(t)\)

In practice, let’s break it down further:

• Identify the outer function (\(f\)) and the inner function (\(g\)). In our example:
  \(f(u) = u^{-4}\) and \(g(t) = 1 + \cos(2t)\).
• Differentiate the outer function \(f(u)\).
• Differentiate the inner function \(g(t)\).
• Multiply the derivatives from these steps.

By applying the chain rule, you combine these individual derivatives to find the complete derivative of the composite function.

The power rule is one of the simplest and most frequently used rules in calculus for differentiating polynomial functions. It states that for a function \(f(x) = x^n\), its derivative is \(f'(x) = nx^{n-1}\). This rule efficiently provides the slope of the tangent line at any point on a curve described by a power function.

In the problem given, the outer function \(u^{-4}\) calls for the application of the power rule. Here’s a quick recap on how it was applied:

• The function is \(u = u^{-4}\).
• Differentiate using the power rule, which gives us: \(\frac{d}{du}u^{-4} = -4u^{-5}\).

Using the power rule, we simplified the process significantly and were able to quickly move on to working with the inner function.

Trigonometric functions are also common in calculus. When differentiating trigonometric functions, it’s essential to remember their specific derivatives. In our specific problem, the function involved \(\cos(2t)\), the derivative of which uses knowledge of both the cosine function and the chain rule.

Here are key derivatives you should remember:

• \(\frac{d}{dt} \cos(t) = -\sin(t)\)
• \(\frac{d}{dt} \sin(t) = \cos(t)\)

In this problem:

• The expression inside the trigonometric function was \(\cos(2t)\), incorporating an inner function \(2t\).
• Using both the chain rule and derivative of cosine, we differentiated the inner function, which yields: \(\frac{d}{dt} \cos(2t) = -2\sin(2t)\).

Differentiating trigonometric functions might seem challenging at first, but by understanding their basic derivatives and using the chain rule when needed, it becomes a methodical process.