27–76. Calculate the derivative of the following functions.
$\sqrt[3]{x^2+9}$

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The function is \( f(x) = \sqrt[3]{x^2 + 9} \).
2. Step 2: Recognize that this is a composition of functions, where the outer function is a cube root and the inner function is \( x^2 + 9 \).
3. Step 3: Apply the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), then its derivative is \( f'(g(x)) \cdot g'(x) \).
4. Step 4: Differentiate the outer function \( u = \sqrt[3]{u} \) with respect to \( u \), which gives \( \frac{1}{3}u^{-2/3} \).
5. Step 5: Differentiate the inner function \( g(x) = x^2 + 9 \) with respect to \( x \), which gives \( 2x \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is essential when dealing with complex functions, such as those involving roots or powers.

Power Rule

The power rule is a basic rule for finding the derivative of a function in the form of x raised to a power. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions and is often used in conjunction with other rules, such as the chain rule.