91Ó°ÊÓ

The chain rule is a crucial tool for differentiating composite functions, meaning functions of the form \( f(g(x)) \). It helps you find the derivative of a function that is nested inside another function. To use the chain rule, you follow these steps:

• Differentiating the outer function while keeping the inner function unchanged.
• Then, multiplying by the derivative of the inner function.

In our exercise, we applied the chain rule when dealing with the term \( \sin(2x)^2 \). Here, the outer function is the square, and the inner function is \( \sin(2x) \). We first differentiated the square, getting \( 2\sin(2x) \). Then, we found the derivative of \( \sin(2x) \), which is \( 2\cos(2x) \). Multiplying these together gives the derivative of the composite function \( \frac{d}{dx}[\sin(2x)^2] = 4\sin(2x)\cos(2x) \).
Understanding and using the chain rule is essential for tackling more complex functions and identifying the relationship between nested functions easily.

Square root differentiation involves differentiating a function where a variable is under a square root sign. If you take a closer look at \( \sqrt{x} \), mathematically it's equivalent to \( x^{1/2} \). Using the power rule for differentiation where \( (x^n)' = nx^{n-1} \), we apply it here with \( n = \frac{1}{2} \).

• Differentiate using the power rule: \( \frac{1}{2} x^{-rac{1}{2}} \).
• This essentially gives you \( \frac{1}{2\sqrt{x}} \).

In the original exercise, differentiating \( \sqrt{x} \) resulted in \( \frac{1}{2\sqrt{x}} \). This step is fundamental as it converts the expression into a form that is easier to work with for further calculations, especially when combined with other derivative parts.

Trigonometric differentiation involves finding derivatives of trig functions such as sine and cosine. These functions have straightforward derivatives that are crucial in calculus:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).

In our specific exercise, we had to differentiate \( \sin(2x) \) as part of the chain rule application. The differentiation of a trigonometric function like \( \sin(2x) \) involves using the chain rule since it is in the form \( \sin(g(x)) \) where \( g(x) = 2x \).
The derivative of \( \sin(2x) \) leads to \( 2\cos(2x) \). This result is then used to find the derivative of any expression involving \( \sin \), such as \( \sin(2x)^2 \), by multiplying with the derivative of the outer function, showcasing how fundamental these roles are in calculus operations involving trig functions.