91Ó°ÊÓ

In calculus, the chain rule is an essential technique when finding the derivative of compositions of functions. Imagine you have a function like a nested Russian doll, where one function is inside another function. Using the chain rule allows you to "unwrap" the outer function and multiply its derivative by the derivative of the inner function.

For the function \( f(x) = ((x^3 + 2)^{1/2} + 2x)^{-2} \), the outer function here can be considered as \((g(x))^{-2}\) and the inner function \( g(x) = (x^3 + 2)^{1/2} + 2x \).

• First, take the derivative of the outer function, treating the inner function as a single unit.
• The derivative of the outer function is \(-2(g(x))^{-3}\).
• Next, multiply this by the derivative of the inner function, \( g'(x) \).

By applying the chain rule, you get the correct derivative involving both inner and outer functions, beautifully stitched together.

Differentiation is the process of finding the derivative of a function. The derivative provides information about the rate of change of the function's output concerning its input. This is a powerful tool in calculus that allows you to understand how the function behaves.

Consider the inner function \( g(x) = (x^3 + 2)^{1/2} + 2x \). To find its derivative, you differentiate each term:

• For \((x^3 + 2)^{1/2}\), use the power rule while treating it as \((x^3 + 2)^{1/2}\). The derivative is \((1/2)(x^3 + 2)^{-1/2} \times 3x^2\).
• The derivative of \(2x\) is simply \(2\).

Combine these to get the derivative of the entire inner function: \( g'(x) = (1/2)(x^3 + 2)^{-1/2} \times 3x^2 + 2 \).

This derivative \( g'(x) \) plays a crucial role when coupled with the chain rule for finding \( f'(x) \).

Simplifying mathematical expressions is vital, especially in calculus, to make derivative calculations more manageable and understandable. Once you have applied the chain rule and found the derivative, it's often filled with complex notations and fractional exponents. Simplifying these can help in gaining clearer insights or further calculations.

For the derivative of the function given, simplification involves:

• Combining like terms, such as similar powers of \((x^3 + 2)\).
• Factoring out common components to make the expression more compact.

The derivative, \( f'(x) = -2((x^3 + 2)^{-1/2} \times 3x^2 + 2) \times ((x^3 + 2)^{1/2} + 2x)^{-3}\), looks complex. However, by carefully grouping terms and eliminating redundancies, it becomes easier to manage. This step is crucial, whether you're looking to interpret the result graphically, using further calculus, or simply ensuring that your work is tidy and accurate.