91Ó°ÊÓ

The power rule is one of the most important concepts in differentiation.

When you see a term like \(x^n\) and you want to find its derivative, the power rule helps you. By applying the power rule, \( \frac{d}{dx} x^n = n x^{n-1} \), you multiply the exponent \(n\) by the coefficient in front of \(x\), and then decrease the exponent by one.

For example, if you have \(5x^3\), using the power rule, the derivative would be \(5 \times 3 \times x^{3-1}\), which simplifies to \(15x^2\). This rule is easy to apply and extremely useful for polynomials and many other functions involving powers of \(x\).

Let's now see this in action with our function \( y=7x^{-3}\). By using the power rule, we multiply \(-3\) by \(7\) (coefficient) to get \(-21\). Then, we reduce the exponent by one (from \(-3\) to \(-4\)), giving us \(-21x^{-4}\).

Negative exponents are simpler than they might seem.

A negative exponent indicates reciprocal. For example, \(x^{-n}\) equals \(\frac {1}{x^n}\). It's useful to turn negative exponents into fractions, especially when simplifying results.

Let's look more closely at our function \( y=\frac{7}{x^{3}} \). By rewriting \(\frac{7}{x^{3}}\), we get \( 7x^{-3}\).

This makes differentiation easier because we can apply the power rule without dealing directly with fractions.

After differentiating \(7x^{-3}\) using the power rule, we got \(-21x^{-4}\). If we wish to express the derivative with positive exponents, we can take \(x^{-4}\) and turn it into the reciprocal of \(x^4\), leading to \(\frac{-21}{x^{4}}\). Both forms are correct, so choose the one that's easier for you to work with.

Simplifying derivatives often involves a few steps:

• Combining like terms
• Rewriting expressions with negative exponents as fractions
• Factoring out common terms

Let's break this down with our derivative \( \frac{d y}{d x} = -21x^{-4}\).

If we wish to simplify it, we convert \(x^{-4}\) to \( \frac{1}{x^4}\). This turns \(-21x^{-4}\) into \( \frac{-21}{x^4}\).

While both \(-21x^{-4}\) and \( \frac{-21}{x^4}\) represent the same quantity, some forms may be more suitable based on the context of your problem. Simplifying the derivative can make further calculations easier and provide a clearer understanding of the function's rate of change.