91Ó°ÊÓ

When it comes to the absolute value function, \(|x|\), the derivative is a bit special. This is because the function itself behaves differently depending on whether \(x\) is positive or negative. Here's how we differentiate an absolute value:

• If \(x > 0\), then \(|x| = x\) and the derivative \(\frac{d}{dx}(x) = 1\).
• If \(x < 0\), then \(|x| = -x\) and the derivative \(\frac{d}{dx}(-x) = -1\).

So, the derivative of \(|x|\) can be summarized as follows:\[\frac{d}{d x}(|x|) = \begin{cases} 1, & x > 0, \-1, & x < 0 \end{cases}\]This format tells us to consider the sign of \(x\) when differentiating. Understanding this principle helps us to tackle more complex absolute value functions, such as \(|\sin x|\), later. It requires the same approach, considering the sign of the inner function \(\sin x\).

Differentiation is a fundamental calculus operation that lets us find the rate at which a function changes. There are several techniques you can use, and sometimes they can be combined:

• Power Rule: Useful for polynomials, given by \(\frac{d}{dx} x^n = nx^{n-1}\).
• Product Rule: For functions that are products; if \(u\) and \(v\) are functions, \(\frac{d}{dx}(uv) = u'v + uv'\).
• Quotient Rule: For functions that are quotients; if \(u\) and \(v\) are functions, \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\).
• Chain Rule: Very important for composite functions, given by \(\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\).

Let's focus on the chain rule, which is central for differentiating nested functions like \(|\sin x|\). Here, the chain rule tells us how to differentiate by breaking down a complex function into simpler parts.First, identify the outer and inner functions. In \(|\sin x|\), the outer function is the absolute value, while the inner function is \(\sin x\). Differentiating them separately and then combining, respecting the signs, leads us to the correct derivative.

Trigonometric functions such as \(\sin(x)\) and \(\cos(x)\) have well-known derivatives that are essential for calculus:

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

When applying these derivatives, especially within other functions due to the chain rule, it's crucial to remember their signs and periodic properties. In our specific problem, we deal with \(|\sin x|\), where you need the derivative of \(\sin x\), which we know is \(\cos x\).By breaking down the function \(|\sin x|\) and using the chain rule, you find that the signs of \(\sin x\) will switch the sign of the derivative. Therefore, when \(\sin x > 0\) (positive sine values), the derivative relies on \(\cos x\), and when \(\sin x < 0\), the derivative is \(-\cos x\). This shows the interplay and importance of both the chain rule and trigonometric derivatives in solving such problems.