Problem 23 - Calculus and its applications

Question

Differentiate each function $$ f(x)=-5 x(2 x-3)^{4} $$

Step-by-Step Solution

Answer

\( f'(x) = -5(2x-3)^4 - 40x(2x-3)^3 \)

Step 1: Identify the Function Components

The given function is \( f(x) = -5x(2x-3)^4 \). It's a product of two parts: \( -5x \) and \( (2x-3)^4 \). This suggests the use of the product rule for differentiation.

Step 2: Recall Product Rule

The product rule states that the derivative of a product \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \). Here, let \( u(x) = -5x \) and \( v(x) = (2x-3)^4 \).

Step 3: Differentiate \( u(x) = -5x \)

Find \( u'(x) \), the derivative of \( u(x) = -5x \). The derivative of \( -5x \) with respect to \( x \) is \( -5 \). So, \( u'(x) = -5 \).

Step 4: Differentiate \( v(x) = (2x-3)^4 \)

Find \( v'(x) \) using the chain rule. First, the derivative of \( (2x-3)^4 \) with respect to \( 2x-3 \) is \( 4(2x-3)^3 \). Then, multiply by the derivative of \( 2x-3 \), which is 2. Thus, \( v'(x) = 4 imes 2 imes (2x-3)^3 = 8(2x-3)^3 \).

Step 5: Apply the Product Rule

Using the product rule, we find that:\[ f'(x) = u'(x)v(x) + u(x)v'(x) = (-5)(2x-3)^4 + (-5x)(8(2x-3)^3) \].

Step 6: Simplify the Expression

Simplify the expression for \( f'(x) \):- The first term is \( -5(2x-3)^4 \).- The second term is \( -5x \cdot 8(2x-3)^3 = -40x(2x-3)^3 \).Combine both terms:\[ f'(x) = -5(2x-3)^4 - 40x(2x-3)^3 \].

Key Concepts

Product Rule

When we're dealing with a function that is the product of two or more simpler functions, the product rule is indispensable. This rule comes in handy for functions like the one we're working with: \[ f(x) = -5x(2x-3)^4 \]. The product rule states that the derivative of a product of two functions, let's call them \( u(x) \) and \( v(x) \), is given by the formula: \[ u'(x) v(x) + u(x) v'(x) \].
In this formula, \( u'(x) \) and \( v'(x) \) represent the derivatives of \( u(x) \) and \( v(x) \) respectively. It's like a team effort where you partially differentiate one function while keeping the other unchanged, and then switch.

• First, differentiate \( u(x) = -5x \), resulting in \( u'(x) = -5 \).
• Next, address \( v(x) = (2x-3)^4 \), which needs a different rule for its derivative.

Chain Rule

The chain rule is crucial when the function you're differentiating is a composition of two or more functions. It's like peeling an onion, where you handle the outer layer first and then the inner layer. In our function \( f(x) = -5x(2x-3)^4 \), the part \( v(x) = (2x-3)^4 \) is a composite function. You have an "inner" function \( 2x-3 \) and an "outer" function \( (u)^4 \) where \( u = 2x-3 \).
Using the chain rule, if \( y = g(u) \) and \( u = h(x) \), then the derivative \( \frac{dy}{dx} \) is computed as:\[ \frac{dy}{du} \times \frac{du}{dx} \].

• Differentiate \( (2x-3)^4 \) with respect to its inner part \( 2x-3 \), which gives \( 4(2x-3)^3 \).
• Then multiply this result by the derivative of the inner function \( 2x-3 \), which is 2, obtaining a full derivative \( 8(2x-3)^3 \).

Combining these steps using the product rule and chain rule allows us to obtain the derivative for the entire function.

Derivative Simplification

After applying the appropriate differentiation rules, we often end up with a complex expression that requires simplification. Simplifying derivatives makes them easier to interpret and use in further calculations. In the context of our function \( f(x) = -5x(2x-3)^4 \), once we apply the product and chain rule, we arrive at: \[ f'(x) = -5(2x-3)^4 + (-5x)(8(2x-3)^3) \].
To simplify, we need to manage these terms carefully:

• The first term remains as it is: \(-5(2x-3)^4\).
• Simplifying the second term involves multiplying: \(-5x \cdot 8(2x-3)^3 = -40x(2x-3)^3\).

Both terms are now neat and manageable, culminating to: \[ f'(x) = -5(2x-3)^4 - 40x(2x-3)^3 \]. This is the fully simplified expression for the derivative of the function, and it provides insight into the behavior of the original function.