91Ó°ÊÓ

In differential calculus, calculating a derivative is like finding how a function changes as its input changes slightly. For the exercise provided, we want to calculate the derivative of the function \( y = \frac{1}{x+1} \). To simplify the process, we can rewrite this as \( y = (x+1)^{-1} \). Calculating the derivative involves using techniques that determine the rate of change. By rewriting the function, we make it easier to apply the power rule (which we'll explain later) instead of dealing with quotients. After applying the method of differentiation, we find that the derivative \( \frac{dy}{dx} \) is \(-\frac{1}{(x+1)^2}\). This means for every tiny change in \(x\), we can calculate how much \(y\) is expected to change, which is exactly what a derivative tells us.

Differential equations involve equations that relate a function with its derivatives. The exercise here only involves basic differential methods, calculating what's called a simple differential \( dy \). A differential like \( dy \) helps us approximate how much \( y \) changes when \( x \) is changed by a small amount, \( dx \). To compute \( dy \):

• We multiply the derivative \( \frac{dy}{dx} \) by the small change in \(x\), \( dx \).
• This yields \( dy = \frac{-1}{(x+1)^2} \cdot dx \).

Thus, given an initial value for \( x \) and a tiny adjustment \( dx \), we can estimate the resultant change in \( y \). Such approximation techniques are useful across many scientific fields when solving problems relating to how variables interact.

One of the key skills in differential calculus is knowledge of differentiation rules, including the power rule. It’s a straightforward method to find the derivative of polynomial-like terms. The power rule states: if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \). In our exercise, the original function \( y = (x+1)^{-1} \) is in a form that allows application of the power rule. By setting the exponent to \(-1\), we can directly find its derivative. Applying the rule, the exponent \(-1\) is multiplied by the original function, reducing the power by one and giving us \( -1(x+1)^{-2} \). Knowing this rule allows us to compute derivatives swiftly without cumbersome algebra like dealing with fractions unless necessary.