91Ó°ÊÓ

When dealing with the differentiation of a product of two functions, the **Product Rule** comes into play. The product rule is essential in calculus when you have a function that is the product of two differentiable functions, say \(u(x)\) and \(v(x)\). To find the derivative of their product, the product rule states:

• \(\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'\)

Here, \(u'\) is the derivative of \(u\) with respect to \(x\), and \(v'\) is the derivative of \(v\) with respect to \(x\).
It’s important to apply this rule correctly by ensuring both functions are differentiated separately and then combined as per the rule.
In our original exercise, we identified \(u(x) = e^{-5x}\) and \(v(x) = \cos 3x\). We found \(u'(x) = -5e^{-5x}\) and \(v'(x) = -3\sin 3x\), then substituted these into the product rule formula to get the derivative of the entire function.

The **Chain Rule** is a fundamental principle used when differentiating composite functions. A composite function is like a function within a function, such as \(f(g(x))\). To differentiate such functions, you apply the chain rule, which can be expressed as:

• \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

This rule allows you to take the derivative of the outer function \(f\), evaluated at \(g(x)\), and then multiply it by the derivative of the inner function \(g(x)\).
Applying this to our original function, we differentiate \(\cos(3x)\). Here, \(f(x) = \cos x\) and \(g(x) = 3x\). Thus, \(f'(x) = -\sin x\), so using the chain rule, we find the derivative is \(-3\sin(3x)\).
This careful application of the chain rule helps manage the complexity and ensures that each function level is considered.

Differentiating trigonometric functions involves recognizing specific derivatives of the sine and cosine functions, among others. The most common derivatives you need to remember for sine and cosine are:

• \(\frac{d}{dx}(\sin x) = \cos x\)
• \(\frac{d}{dx}(\cos x) = -\sin x\)

These are crucial when dealing with problems involving trigonometric differentiation.
In our case, the function \(\cos 3x\) required differentiation. By applying the chain rule combined with these trigonometric derivatives, we achieved an accurate result. Noticing these patterns and derivatives helps you anticipate what the outcome should be.
This approach simplifies the complexity of differentiation problems involving trigonometric functions, making it more manageable to combine rules like the product rule together with trigonometric differentiation techniques.