91Ó°ÊÓ

The power rule is essential for differentiating functions of the form \( y = x^n \). By applying the power rule, you can quickly find the derivative. The general form says if you have \( y = x^n \), then the derivative \( y' \) is calculated as \( \frac{d}{dx} x^n = n x^{n-1} \).

For example, if \( y = x^4 \), applying the power rule gives \( y' = 4x^3 \).

Why is this useful? When you rewrite more complex functions into a form like \( x^n \), differentiating becomes straightforward. You then adjust the exponent and multiply by the original exponent, stepping down the power by one. For instance, with \( y = x^{1/3} \), the power rule transforms this into \( \frac{1}{3} x^{-2/3} \).

Simple, right?

The constant multiple rule helps when your function includes a constant multiplier. This rule states that if you have a function \( y = c * f(x) \), where \( c \) is a constant, then its derivative \( y' \) is \( c \times f'(x) \).

For example, consider \( y = 5x^2 \). The derivative is found by keeping the constant 5 and differentiating \( x^2 \) using the power rule: \( 5 \times 2x = 10 x \).

In our exercise, we have \( y = 3 x^{1/3} \). Applying the constant multiple rule involves keeping the constant 3 while differentiating \( x^{1/3} \) using the power rule. This gives \( y' = 3 \times \frac{1}{3} x^{-2/3} \).

Simplified, this becomes \( y' = x^{-2/3} \). Easy to manage, once you understand the rules involved!

Understanding cube roots can make differentiation tasks simpler. The cube root of a number \( x \), written as \( \root{3}{x} \), is the value that, when cubed, gives \( x \). For example, \( \root{3}{27} = 3 \) and \( 3^3 = 27 \).

For differentiating, it's easier to convert a cube root into an exponent form. A cube root is effectively the exponent \( 1/3 \). So, \( \root{3}{x} \) can be written as \( x^{1/3} \).

Once written in exponent form, leveraging differentiation techniques like the power rule becomes straightforward.

In our exercise, rewriting \( 3 \root{3}{x} \) as \( 3 x^{1/3} \) transforms the problem into a simple differentiation task. You then use known rules like the power rule and constant multiple rule to find the derivative. This structured method helps tackle more complex-looking functions effortlessly.