91Ó°ÊÓ

In differential calculus, a *first derivative* provides important information about the rate of change of a function. Essentially, it tells us how a function's output value changes with respect to its input value. For the function given in our exercise, which is \( y = x + \tan x \), we need to find the first derivative to understand this change.

The first derivative of a function, denoted as \( y' \), is obtained by differentiating the function with respect to \( x \). Here, we apply the sum rule, which states that the derivative of a sum of two functions is the sum of their derivatives.

• The derivative of \( x \) is simply \( 1 \), because it increases by one unit as \( x \) increases by one unit.
• The derivative of \( \tan x \) is \( \sec^2 x \), a fundamental trigonometric derivative.

So, combining these results, we find that the first derivative of \( y = x + \tan x \) is \( y' = 1 + \sec^2 x \). This result gives us the rate of change of \( y \) with respect to \( x \).

The *second derivative* provides insight into the curvature of a function. It indicates how the rate of change of a function's derivative itself changes. In other words, it tells us about the acceleration or deceleration of our original function.

For our function \( y = x + \tan x \), and its first derivative \( y' = 1 + \sec^2 x \), we now need to find the second derivative. This involves differentiating \( y' \) with respect to \( x \).

• \( \sec^2 x \) can be rewritten as \( \frac{1}{\cos^2 x} \) which helps when applying the chain rule.
• The second derivative of a constant term, \( 1 \), is zero.
• For \( \sec^2 x \), using chain rule, the derivative involves \( 2\sec^2 x\tan x \), resulting from differentiating \( \sec^2 x \) with respect to \( x \).

Therefore, after applying these concepts, the second derivative is \( \frac{d^2 y}{dx^2} = 2\tan x / \cos^2 x \). This measures how \( y' \), and hence \( y \), changes over time.

The *chain rule* is a fundamental method in differential calculus used to differentiate compositions of functions. It can be incredibly useful when finding derivatives of complex functions, such as in our second derivative calculation.

In essence, the chain rule allows us to understand the derivative of a "chain" of functions, where one function is nested inside another. Mathematically, if you have functions \( f(g(x)) \), then their derivative would be \( f'(g(x)) \cdot g'(x) \).

For instance, in our exercise, we need to apply the chain rule to \( \sec^2 x \), because \( \sec x \) can be seen as an outer function applied to the angle expressed through \( x \).

• Identify the outer function \( f(u) = u^2 \) where \( u = \sec x \).
• The inner function, \( g(x) = \sec x \) with its derivative being \( \sec x \tan x \).
• Apply the chain rule: \( \frac{d}{dx}(\sec^2 x) = 2\sec x \cdot (\sec x \tan x) = 2\sec^2 x \tan x \).

This rule simplifies finding complex derivatives gracefully, as it guides us through breaking down the differentiation into manageable parts.