Question

Find the derivative. $$\frac{d}{d x}\left(1.5 x^{2}-3\right)^{3}$$

Short answer

\(\frac{d}{dx}\left(1.5x^2 - 3\right)^3 = 13.5x \cdot (1.5x^2 - 3)^2\)

Step-by-step solution

Identify the Outer Function

Recognize that the given function is a composition of two functions, with the outer function being raised to a power of 3. This implies we will need to use the chain rule.

Apply the Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Let's denote the inner function as \(u(x) = 1.5x^2 - 3\). Then, the derivative of the outer function \(f(u) = u^3\) with respect to \(u\) is \(f'(u) = 3u^2\).

Differentiate the Inner Function

Now we need to find the derivative of the inner function \(u(x) = 1.5x^2 - 3\). The derivative is \(u'(x) = 3 \cdot 1.5x = 4.5x\).

Combine Derivatives Using the Chain Rule

Combining the derivatives according to the chain rule results in the overall derivative: \(\frac{d}{dx}\left(1.5x^2 - 3\right)^3 = f'(u) \times u'(x) = 3u^2 \times 4.5x = 3(1.5x^2 - 3)^2 \times 4.5x\).

Simplify the Derivative

Finally, we can simplify the expression by distributing and multiplying all constants and variables together: \(\frac{d}{dx}\left(1.5x^2 - 3\right)^3 = 13.5x \cdot (1.5x^2 - 3)^2\).

Key concepts

Composite Function

When faced with problems in calculus that involve multiple functions interacting with one another, we encounter what is known as a composite function. This is essentially a function made up of two or more functions, where the output of one function becomes the input for another.

Imagine a composite function as a machine with multiple stages. The input, or independent variable, goes through the first stage - the inner function - which then processes it and passes it along to the next stage, the outer function. In the exercise, we have an inner function given by \( u(x) = 1.5x^2 - 3 \), which is then fed into the outer function \( f(u) = u^3 \) . Understanding the composition is critical because it forms the basis for successfully applying the chain rule for differentiation.

Derivative Calculation

The process of finding the derivative of a function is what we refer to as derivative calculation. It's a fundamental tool in calculus that measures the rate at which a quantity changes. When you're told to 'find the derivative', you're essentially being asked to find the slope of the function at any point, or the instantaneous rate of change.

Let's focus on the provided problem, where the derivative is found using a step-by-step approach. This methodical technique is crucial for managing more complex expressions. It starts by recognizing the function's structure and then progressively applying appropriate rules of differentiation, in this case, the chain rule. Remember, any time you're working with a composite function, understanding the separate pieces is essential for accurate derivative calculation.

Differentiation Techniques

Mastering differentiation techniques is key to success in calculus. Different situations call for different methods, and capacity to choose the right tool can streamline the calculation process. The chain rule is just one of these essential techniques, but there are others like the product rule, quotient rule, and the power rule.

In the given exercise, we employed the chain rule which is perfect for tackling composite functions. The technique involves taking the derivative of the outer function, while keeping the inner function as is, then multiplying by the derivative of the inner function. This step-by-step solution exemplifies a strategic approach, breaking down the process into manageable parts and ensuring each rule is correctly applied. A firm grasp of these differentiation techniques can turn a seemingly intimidating problem into a series of simple calculations.