Problem 18 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ g(x)=(2 x-9)^{2}\left(x^{3}+4 x-5\right)^{3} $$

Step-by-Step Solution

Answer

[4(2x - 9)(x^3 + 4x - 5)^3] + [(2x - 9)^2 3(x^3 + 4x - 5)^2 (3x^2 + 4)]

Step 1: Identify the functions

The given function is a product of two functions. Identify them separately: \(u(x) = (2x - 9)^2\) and \(v(x) = (x^3 + 4x - 5)^3\).

Step 2: Apply the product rule

To find the derivative of the product of two functions, use the product rule: \((u \times v)' = u' \times v + u \times v'\).

Step 3: Find the derivative of \(u(x)\)

To find \(u'(x)\), use the chain rule: \(u(x) = (2x - 9)^2\). Let \(w = 2x - 9\), then \(u(w) = w^2\). So, \(u'(w) = 2w\) and \(w' = 2\). Thus, \(u'(x) = 2(2x - 9) \times 2 = 4(2x - 9)\).

Step 4: Find the derivative of \(v(x)\)

To find \(v'(x)\), use the chain rule: \(v(x) = (x^3 + 4x - 5)^3\). Let \(z = x^3 + 4x - 5\), then \(v(z) = z^3\). So, \(v'(z) = 3z^2\) and \(z' = 3x^2 + 4\). Thus, \(v'(x) = 3(x^3 + 4x - 5)^2 \times (3x^2 + 4)\).

Step 5: Substitute and simplify

Using the product rule, substitute the derivatives found: \((u \times v)' = [4(2x - 9) \times (x^3 + 4x - 5)^3] + [(2x - 9)^2 \times 3(x^3 + 4x - 5)^2 \times (3x^2 + 4)]\). Simplify the expression for the final answer.

Key Concepts

Product Rule

When differentiating a product of two functions, we need to use the product rule. The product rule states that if you have two functions, say \( u(x) \) and \( v(x) \), their derivative is given by: \[ (u \times v)' = u' \times v + u \times v' \]This means we need to find the derivative of the first function, \( u(x) \), multiply it by the undifferentiated second function, \( v(x) \), and then add the product of the undifferentiated first function and the derivative of the second function. This rule is essential when dealing with the multiplication of any two functions. In our example, \(g(x)=(2 x-9)^{2}\big(x^{3}+4 x-5\big)^{3} \), we identified the two functions as \( u(x) = (2x - 9)^2 \) and \( v(x) = (x^3 + 4x - 5)^3 \). Next, we apply the product rule to these functions.

Chain Rule

The chain rule is used to differentiate composite functions. When you have a function nested within another function, the chain rule helps in breaking down the differentiation process into manageable parts. The chain rule states: \[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \]In our example, to differentiate \( u(x) = (2x - 9)^2 \) and \( v(x) = (x^3 + 4x - 5)^3 \), we use the chain rule. For \( u(x) \), we let \( w = 2x - 9 \), then \( u(w) = w^2 \). Differentiating, we find \[ u'(w) = 2w \] and \[ w' = 2 \]. Combining these, we get \( u'(x) = 2(2x - 9) \times 2 = 4(2x - 9) \).For \( v(x) \), we let \( z = x^3 + 4x - 5 \), then \( v(z) = z^3 \). Differentiating, we find \[ v'(z) = 3z^2 \] and \[ z' = 3x^2 + 4 \]. Combining these, we get \( v'(x) = 3(x^3 + 4x - 5)^2 \times (3x^2 + 4) \).

Differentiation Steps

To find the derivative of a given function, follow these differentiation steps:

• Identify the individual functions that make up the product.
• Apply the product rule to differentiate the product of the functions.
• Use the chain rule to manage any composite functions.
• Substitute the derivatives back into the product rule and simplify.

With our function \( g(x) \), we've identified that \( g(x) = (2 x-9)^{2}(x^{3}+4 x-5)^{3} \) is a product of two functions.Using the product rule and our earlier chain rule results, we substitute and get: \[ (u \times v)' = [4(2x - 9) \times (x^3 + 4x - 5)^3] + [(2x - 9)^2 \times 3(x^3 + 4x - 5)^2 \times (3x^2 + 4)] \]Finally, simplify the entire expression to get the derivative of the given function.