91Ó°ÊÓ

Understanding the Chain Rule is essential for differentiating functions, especially those involving composite functions. The Chain Rule lets us differentiate complex compositions by breaking them into simpler parts. Here’s what the rule says:

If you have a composite function like \( y = f(g(x)) \), the Chain Rule formula is:\[ \frac{dy}{dx} = f'(g(x)) \times g'(x) \]To use the Chain Rule:

• Identify the outer function and the inner function.
• Differentiate the outer function with respect to the inner function.
• Differentiate the inner function with respect to \( x \).
• Multiply the two derivatives together.

In our exercise, we have \( y = 3 (2x^2 + 1)^{1/3} \). Here, the outer function is \( (2x^2 + 1)^{1/3} \) and the inner function is \( 2x^2 + 1 \). You’ll first differentiate the outer function and then multiply it by the derivative of the inner function, applying the Chain Rule effectively.

Fractional exponents can seem intimidating, but they are straightforward once you understand them. A fractional exponent is another way of expressing a root. For example, the cube root of \( 2x^2 + 1 \) can be written as \( (2x^2 + 1)^{1/3} \). Some important points to consider:

• \( a^{1/n} \) is the same as the \( n \)th root of \( a \).
• When differentiating a function with a fractional exponent, you'll apply the Chain Rule.

Using the power rule for differentiation, \( (x^n)' = nx^{n-1} \), we accommodate fractional exponents by treating \( n \) as a fraction. For our problem, \( (2x^2 + 1)^{1/3} \) gets differentiated as \( \frac{1}{3}(2x^2 + 1)^{-2/3} \), using the same principles as integer exponents, but making sure to adjust for the fractional power.

Determining which part of a composite function is the inner function and which is the outer function is critical when using the Chain Rule. In our example, we have \( y = 3(2x^2 + 1)^{1/3} \). To break this down:

• Inner function: \( u = 2x^2 + 1 \)
• Outer function: \( y = 3u^{1/3} \)

First, differentiate the outer function with respect to the inner function:

Since \( y = 3u^{1/3} \), we get \( \frac{dy}{du} = 3 \times \frac{1}{3} u^{-2/3} \). After simplifying, we have \( \frac{dy}{du} = u^{-2/3} \).

Next, differentiate the inner function with respect to \( x \):
\( \frac{du}{dx} = \frac{d}{dx}[2x^2 + 1] = 4x \).

Finally, apply the Chain Rule by multiplying these results together:
\( \frac{dy}{dx} = (2x^2 + 1)^{-2/3} \times 4x \).
It simplifies to \( \frac{dy}{dx} = 4x(2x^2 + 1)^{-2/3} \).