91Ó°ÊÓ

The Power Rule is a fundamental tool in calculus used to take the derivative of functions of the form \(y = ax^n\). When using the Power Rule, remember the formula: \[ \frac{d y}{d x} = a \times n \times x^{n-1} \] Here, 'a' is a constant coefficient, and 'n' is the exponent of x.
For example, if your function is \(y = \frac{1}{2} x^{\frac{4}{5}}\), 'a' is \( \frac{1}{2} \) and 'n' is \( \frac{4}{5} \).
The Power Rule helps in transforming the original function into its derivative, making it essential for solving various calculus problems.

After applying the Power Rule, you often end up with an expression that needs simplification. Simplifying ensures your derivative is in its most understandable form.
In the exercise, the initial step of simplification was to multiply the constants: \( \frac{1}{2} \times \frac{4}{5} = \frac{4}{10} = \frac{2}{5} \).
Next, simplifying the exponents is crucial. Subtract 1 from \( \frac{4}{5} \): \( \frac{4}{5} - 1 = \frac{4}{5} - \frac{5}{5} = -\frac{1}{5} \).
By carefully performing these steps, you make the derivative clear and correct: \[ \frac{d y}{d x} = \frac{2}{5} x^{-\frac{1}{5}} \]

Following a step-by-step approach is vital to accurately finding derivatives. First, understand the problem and identify the given function.
Next, recall the Power Rule and identify the parameters ('a' and 'n'). For example, in the given function \(y = \frac{1}{2} x^{\frac{4}{5}}\), 'a' is \( \frac{1}{2} \) and 'n' is \( \frac{4}{5} \).
Then, apply the Power Rule: \( \frac{d y}{d x} = \frac{1}{2} \times \frac{4}{5} \times x^{\frac{4}{5}-1} \).
Finally, simplify the expression by evaluating the constants and the new exponent, leading to the final derivative: \( \frac{d y}{d x} = \frac{2}{5} x^{-\frac{1}{5}} \).
By breaking down the process, you can manage complex problems with confidence.