91Ó°ÊÓ

Understanding how to use the power rule is essential when differentiating functions. The power rule states that if you have a term in the form of \(x^n\), where n is a constant, the derivative of that term is given by \(\frac{d}{dx} x^n = nx^{n-1}\). Let's apply this to each term individually in the function. For instance, if you have \(x^{5/2}\), using the power rule, the first derivative is \(\frac{d}{dx} x^{5/2} = \frac{5}{2} x^{3/2}\). This same rule can be applied to other terms: \(\frac{d}{dx} x^{3/2} = \frac{3}{2} x^{1/2}\) and \(\frac{d}{dx} (-x^{1/2}) = -\frac{1}{2} x^{-1/2}\). Make sure to multiply the exponent by the coefficient and then reduce the exponent by one. This rule is foundational for finding both the first and second derivatives in calculus problems.

The first derivative represents the rate of change or the slope of the original function. For the function given, \(y=x^{5 / 2}+x^{3 / 2}-x^{1 / 2}\), the first derivative was found by applying the power rule to each term: \(\frac{dy}{dx} = \frac{5}{2} x^{3/2} + \frac{3}{2} x^{1/2} - \frac{1}{2} x^{-1/2}\). Each term in the derivative corresponds to the rate of change of the respective term in the original function. This first derivative is crucial as it serves as the base for finding higher-order derivatives, like the second derivative.

After differentiating, it's important to simplify the resulting expressions. Simplification involves combining like terms and ensuring that the expression is as straightforward as possible. For example, in our case with the first derivative \( \frac{5}{2} x^{3/2} + \frac{3}{2} x^{1/2} - \frac{1}{2} x^{-1/2}\), there isn't much to combine, but clearly writing this step helps avoid confusion when performing additional operations. The more simplified and neat your expression is, the easier it will be to find the second derivative. Simplification therefore makes subsequent calculations more manageable and reduces the possibility of errors.