91Ó°ÊÓ

Implicit differentiation is used when you have an equation involving both x and y, and you need to find the derivative, \frac{dy}{dx}\. In some cases, y is not isolated on one side of the equation, hence the need for implicit differentiation. Here's how it's done:

1. Differentiate both sides of the equation with respect to x. Keep in mind that y is also a function of x.

2. Apply standard differentiation rules like the chain rule to differentiate terms involving y.

In the given problem, the equation is \( x^2 - y^2 = 16 \). When differentiating both sides with respect to x, treat y as a function of x. This requires employing the chain rule for differentiating \( y^2 \).

In mathematics, many functions are not given explicitly as \( y = f(x) \). Instead, they may be given implicitly as some relation between x and y. Implicit differentiation allows us to find the derivative in such cases.

Finally, isolating \( \frac{dy}{dx} \) by algebraic manipulation gives us the result.

The chain rule is a fundamental concept in calculus used to differentiate compositions of functions. It states that if a variable y depends on a variable u, which in turn depends on x, then y also depends on x, and the derivative of y with respect to x can be found using:
\[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \].

In the equation from the exercise, for the term \( y^2 \), y is a function of x. By the chain rule, the derivative of \( y^2 \) with respect to x is:
\[ \frac{d}{dx} (y^2) = 2y \frac{dy}{dx} \].

This step is crucial because it allows us to differentiate terms where y is implicitly a function of x. The chain rule was applied in this problem to correctly handle the differentiation of \( y^2 \) and obtain the relationship between x and y's derivatives.

Differentiation is the process of finding the derivative of a function. The derivative, denoted as \( \frac{dy}{dx} \), measures the rate at which a function is changing at any point. It gives us the slope of the tangent line to the curve of the function at any given point.

Key points about differentiation:

• The power rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• The sum rule: The derivative of a sum of functions is the sum of their derivatives.
• The difference rule: The derivative of a difference of functions is the difference of their derivatives.

In the given exercise:

1. To differentiate \( x^2 \) w.r.t x, we use the power rule: \( \frac{d}{dx} (x^2) = 2x \).
2. To differentiate \( -y^2 \), we need the chain rule, which we discussed earlier. The term differentiates to \( -2y \frac{dy}{dx} \), incorporating both the chain rule and the power rule.

By combining these rules, we can differentiate complex equations and relations. The final simplification \( x = y \frac{dy}{dx} \) isolates \( \frac{dy}{dx} \), giving us the derivative of y with respect to x.