Question

Find the derivative. $$D\left(3 x^{3}-5\right)^{3}$$

Short answer

\(27x^2(3x^3-5)^2\)

Step-by-step solution

Identify the Outer Function

Recognize that the given function is a composition of functions. Identify the outer function to be the cubic power function, \(u^3\), where \(u=3x^3-5\).

Differentiate the Outer Function with respect to the Inner Function

Apply the chain rule. The derivative of the outer function with respect to the inner function \(u\) is \(3u^2\). Replace \(u\) with \(3x^3-5\) to find \(3(3x^3-5)^2\).

Differentiate the Inner Function with respect to \(x\)

The inner function is \(3x^3-5\). Its derivative with respect to \(x\) is \(9x^2\).

Apply the Chain Rule

Multiply the derivative of the outer function with respect to the inner function by the derivative of the inner function with respect to \(x\) to find the overall derivative: \(3(3x^3-5)^2\times9x^2\).

Simplify

Combine the constants and simplify the expression to get \(27x^2(3x^3-5)^2\).

Key concepts

Derivative of Composition of Functions

Understanding the derivative of a composition of functions is essential when dealing with complex expressions in calculus. This process is succinctly captured by the chain rule, a powerful tool for computing the derivative of a composite function. Let's break it down:
The chain rule states that if you have a composite function, say, f(g(x)), then the derivative of this composite function can be found by taking the derivative of the outer function (f(u), with u = g(x)) and multiplying it by the derivative of the inner function (g(x)). It's a bit like peeling an onion, layer by layer.
In the context of the provided exercise, the composite function is \( (3x^3 - 5)^3 \) and it consists of an outer function \( u^3 \) and an inner function \( u = 3x^3 - 5 \). Step by step, you first differentiate the outer function with respect to u, and then multiply that by the derivative of the inner function with respect to x. It's a sequential process which ensures you cover each 'layer' of the function for a complete derivative.

Power Rule in Differentiation

The power rule is another fundamental concept in the world of differentiation. It simplifies the process of finding derivatives for functions where the variable is raised to a power. The general formula is straightforward: for any function \( x^n \), its derivative is \( nx^{(n-1)} \). This rule is a real timesaver!
When applying the power rule, always remember to multiply the exponent of x by the coefficient in front of x, and then decrease the exponent by one. It's as if you're reducing its 'power' by one level. In the example \( D(3x^3 - 5)^3 \), we see that the power rule is used in step 3 when differentiating the inner function \(3x^3-5\).
Additionally, the power rule plays a crucial role in step 2. When differentiating the outer function \(u^3\), the power rule tells us that the result is \(3u^2\). So, remember, whenever you're faced with a term to differentiate and it has an exponent, the power rule is your go-to strategy.

Simplifying Derivatives

Lastly, simplifying derivatives is the final puzzle piece. It's all about making your answer cleaner and more understandable, which often helps in further calculations. After applying the chain rule and the power rule, you get to the simplification phase. In this phase, you combine like terms, multiply constants, and organize the expression to its most elegant form.
In the exercise, you'll notice after computing the derivative using the chain rule (steps 1 to 4), you'll end with a rather bulky expression. The simplification in step 5, \(27x^2(3x^3-5)^2\), tidies everything up. Simplifying makes it easier to evaluate the derivative at specific points or to proceed with other operations, such as integrating or graphing.
Here's an improvement tip: always keep an eye out for common factors and opportunities to apply standard algebraic identities when simplifying. This can further reduce complex expressions and save you from potential computation errors.