91Ó°ÊÓ

When dealing with the derivatives of functions that are products of two or more other functions, the product rule is a very useful tool. The product rule is essentially a method for differentiating products of two functions. Let's break it down:

• If you have two functions, say \( u(x) \) and \( v(x) \), and you want to find the derivative of their product \( f(x) = u(x)v(x) \),
• The product rule states: \[ f'(x) = u(x)v'(x) + u'(x)v(x) \]

This simply means you take the derivative of the first function and multiply it by the second function as it is, and then take the derivative of the second function and multiply it by the first function.
In the given exercise, our functions are \( u(x) = e^{x} \) and \( v(x) = \sin x \). Applying the product rule, we found:
\[ f'(x) = e^{x}\cos x + e^{x}\sin x \]
This demonstrates how the product rule combines the rates of change of the two functions.

Trigonometric functions are a key component in many calculus problems, especially those involving oscillatory behavior like sine and cosine. Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).

• The sine function, \( \sin x \), has a derivative of \( \cos x \).
• The cosine function, \( \cos x \), has a derivative of \( -\sin x \).

These properties are critical when solving differentiation problems. They help you understand how the angle change rate affects the function's output.
In our given function \( f(x) = e^{x}\sin x \), recognizing that \( \sin x \) differentiates to \( \cos x \) allows us to correctly apply the product rule for differentiation.
Properly understanding these derivatives is essential for finding solutions accurately.

The exponential function \( e^{x} \) is one of the most important functions in calculus due to its unique properties. In differentiation, it stands out because its derivative is itself; that is, \( \frac{d}{dx} (e^{x}) = e^{x} \).

• This is a unique feature that no other function shares, making exponential functions particularly simple to differentiate.
• The exponential rate of change remains consistent through the differentiation process.

In our exercise, applying this property helps directly while using the product rule. Since our \( u(x) = e^{x} \), its derivative \( u'(x) = e^{x} \) plays a vital role in simplifying the problem.
Using exponential functions alongside trigonometric functions further allows us to leverage the strengths of both, using straightforward calculations to achieve effective solutions.