91Ó°ÊÓ

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. A composite function is essentially a function within another function. In simpler terms, if you have a function defined as \(y = f(g(x))\), the chain rule helps differentiate this by focusing on two things:

• The derivative of the outer function, \(f\), evaluated at the inner function, \(g(x)\).
• The derivative of the inner function, \(g(x)\), with respect to \(x\).

The chain rule formula can be expressed as \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \].
Using this rule allows us to find derivatives of complex functions step by step. In the given exercise, the general framework is to first identify the outer function and treat the inner part as a single unit. Then, find the derivative of that inner part separately. Finally, combine them using multiplication to find the derivative of the entire function.

When dealing with functions that are multiplied together, the product rule is essential. This rule states how to differentiate products of two or more functions. For two functions \(u(x)\) and \(v(x)\), the product rule is given by:

• The first function's derivative multiplied by the second function unchanged.
• Plus the first function unchanged multiplied by the second function's derivative.

Mathematically, this is described as \[ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \].
In our specific exercise, we apply the product rule to differentiate the term \(x \sin 2x\).
Here, \(x\) is the first function and \(\sin 2x\) is the second.
Differentiating each part separately, you then add them according to the product rule to find the derivative of the whole product part effectively.

Trigonometric differentiation involves finding derivatives of trigonometric functions. These functions include \(\sin\), \(\cos\), \(\tan\), and their derivatives are essential for solving many calculus problems.
For basic trigonometric functions, the derivatives are as follows:

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

In the problem, we encounter \(\tan^4(x^7)\) where advanced differentiation techniques such as the power rule and chain rule were incorporated. The derivative \(\frac{d}{dx}[\tan^4(x^7)]\) involves understanding that \(\tan^4(x^7)\) is \((\tan(x^7))^4\).
Here, by applying a combination of chain and power rules, the differentiation is simplified into manageable parts, yielding a clear and concise derivative function.