91Ó°ÊÓ

When differentiating a function, we often encounter products of two functions. In such cases, the **product rule** is a handy tool. It's a basic formula used in differentiation, which helps when a function is composed of two multiplied variables, each dependent on the differentiation variable.
Imagine you have a function given as the product of two functions, let's name them: \( u(x) \) and \( v(x) \). The product rule states that the derivative of their product with respect to \( x \) is:

• \( \frac{\mathrm{d}}{\mathrm{d}x} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

In the context of the problem given, where you have \( x \tan y \) on the left and \( y \sin x \) on the right, both expressions consist of products. Therefore, using the product rule helps efficiently handle the differentiation of these expressions.
Always remember:

• Keep one function constant while differentiating the other, then switch their roles.
• Add the results to get the total derivative of the product.

The world of calculus frequently involves **trigonometric functions**. They're foundational in many mathematical problems, including our current differentiation task. Familiarizing oneself with derivatives of basic trigonometric functions can make a significant difference in solving such problems smoothly.
Here's a quick review of the derivatives of basic trigonometric functions:

• 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 original exercise, knowing the derivatives of \( \sin x \) and \( \tan y \) is crucial. For instance, differentiating \( \tan y \) with respect to \( x \) involves understanding the chain rule, which we'll discuss next. Using trigonometric identities and derivatives effectively allows for successful handling of partial derivatives in complex formulas.
Practicing these derivatives and understanding how to apply them will simplify differentiation tasks with trigonometric functions involved.

In differentiation, the **chain rule** is extremely useful for solving functions nested within other functions. It's especially pivotal when dealing with composite functions where one function sits "inside" another.
For example, you have \( \tan y \) in the given problem, where \( y \) itself is a function of \( x \). This is where the chain rule comes into play. If \( y = f(x) \), then when finding the derivative of \( \tan y \) with respect to \( x \), you chain the derivative of the outside function (\( \tan y \)) with the inside function's derivative (\( y \)).
The chain rule formula is:

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

This rule is a lifeline in ensuring you cover all parts of a derivative fully without missing how changes in one variable affect another.
For instance, the derivative of \( \tan y \) with respect to \( x \) is \( \sec^2 y \cdot \frac{\mathrm{d}y}{\mathrm{d}x} \). You first differentiate \( \tan y \) as if \( y \) were a simple variable, and then multiply by the derivative of \( y \) with respect to \( x \).
Mastering the chain rule will make working with partially differentiated equations much easier and more intuitive.