Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(x)=x(2 x-5)^{3} $$

Short answer

The derivative of the function \(f(x) = x(2x - 5)^3\) is \(f'(x) = (2x - 5)^3 + 6x(2x - 5)^2.\)

Step-by-step solution

Identify the functions

Identify the functions that will be differentiated using the product rule. Name the first function as \(u(x) = x\) and the second function as \(v(x) = (2x - 5)^3\)

Apply the Product Rule

First, differentiate \(u(x)\) with respect to x to get \(u'(x) = 1\). Leave \(v(x)\) as is for now. Then, leave \(u(x)\) as is, and differentiate \(v(x)\) with respect to x. For differentiating \(v(x)\), you'll need to apply the chain rule, which leads us to the next step.

Apply the Chain Rule

To differentiate \(v(x) = (2x - 5)^3\), let's make \(2x - 5\) the inner function. It needs to be differentiated as well since it is not a constant. Differentiate the outer function to get \(3(2x - 5)^2\) and multiply it by the derivative of \(2x - 5\), which is 2. So, \(v'(x) = 2 * 3 * (2x - 5)^2 = 6(2x - 5)^2\)

Apply the Product Rule Fully

Use both \(u'(x)\) and \(v'(x)\) from Steps 2 and 3 to fully apply the product rule. This equates to \(f'(x) = u'(x)v(x) + u(x)v'(x) = [(1)(2x - 5)^3] + [x * 6(2x - 5)^2] = (2x - 5)^3 + 6x(2x - 5)^2\).

Key concepts

Product Rule

The product rule is a vital concept in calculus, particularly when dealing with the differentiation of functions that are products of two or more other functions. When a function, say f(x), can be expressed as the product of two sub-functions, like f(x) = u(x)v(x), we can't simply differentiate each part independently and multiply the results. Instead, we use the product rule.

The product rule states that the derivative of u(x)v(x) is given by u'(x)v(x) + u(x)v'(x), where u'(x) and v'(x) are the derivatives of u(x) and v(x), respectively. This rule is essential because it reflects how changes in one function impact the other and vice versa. When applying the product rule, it’s important to differentiate each function separately before combining the results as per the rule.

Example: If you have the function f(x) = x(2x - 5)^3, the first function u(x) is x and the second v(x) is (2x - 5)^3. Here, using the product rule is crucial to finding the derivative correctly. The aim is to break down complex expressions into simpler parts that are easier to manage, which can be especially helpful when functions are more compound or intricate.

Chain Rule

The chain rule comes into play when you need to differentiate a composite function, sometimes thought of as a function within another function. It's like peeling an onion, where you first differentiate the outer layer, and then go deeper to differentiate the inner layers.

Mathematically, if g is a function that is composed of another function f, such that g(x) = h(f(x)), then the derivative of g with respect to x is g'(x) = h'(f(x)) * f'(x). This means you first differentiate the outer function h as if the inner function f was a simple variable, and then multiply the result by the derivative of the inner function f.

In the context of our example, v(x) = (2x - 5)^3 is the composite function to differentiate. The inner function is 2x - 5, and the outer function is something raised to the third power. According to the chain rule, you would differentiate the outer function first, which gives you 3(2x - 5)^2, and then multiply by the derivative of the inner function, which is 2. This yields v'(x) = 6(2x - 5)^2, completing the chain rule process.

Differentiation Steps

Understanding the step-by-step process of differentiation is essential to tackle any calculus problem methodically and accurately. Differentiation can sometimes be straightforward, but it often requires careful application of rules like the product rule and chain rule, among others.

Here's how you'd typically approach differentiation:

• Identify the type of function you're dealing with (simple, product, quotient, composite, etc.)
• Decide which differentiation rules (product, quotient, chain, etc.) are applicable.
• Apply the relevant rule(s) step by step, making sure to adhere closely to the formulae.
• If necessary, simplify the resulting expression to make it easier to understand or further manipulate.

For the given function f(x) = x(2x - 5)^3, one would follow these differentiation steps to successfully find the derivative. Identify x as u(x) and (2x - 5)^3 as v(x), then apply the product rule by first finding the derivatives of u(x) and v(x) separately. With v(x), employ the chain rule, and finally, combine the derivatives as specified by the product rule to obtain f'(x). Proper execution of these steps leads to a clear, structured approach to differentiation, which is advantageous for complex functions.