Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=x(3 x-9)^{3} $$

Step-by-Step Solution

Answer

\(f'(x)=9x(3x-9)^2 + (3x-9)^3\)

Step 1: Identify the First Function (u) and the Second Function (v)

For the Product Rule, the given function needs to be identified as \(u(x)v(x)\). Here, \(u(x)=x\) and \(v(x)=(3x-9)^3\)

Step 2: Differentiate u and v

Differentiate \(u(x)\) and \(v(x)\) separately. The derivative of \(u(x)\) is \(u'(x)=1\). To find \(v'(x)\), use the Chain Rule, since \(v(x)\) is a composite function. Identify the outer function as \((h(x))^3\) and the inner function as \(h(x)=3x-9\). Then, differentiate the outer function using the Power Rule, and the inner function using the Constant Multiple Rule. The derivative of the outer function is \(3(h(x))^2\), and the derivative of the inner function is 3. By the Chain Rule, the derivative of the composite function is the product of these, which is \(v'(x)=9(3x-9)^2\).

Step 3: Apply the Product Rule

The Product Rule states that the derivative of \(u(x)v(x)\) is \(u(x)v'(x) + v(x)u'(x)\). So, substituting the calculated derivatives and functions we get \(f'(x)=x*9(3x-9)^2 + (3x-9)^3*1\)

Step 4: Simplify the Expression

Simplify the derivative expression to get the final answer, \(f'(x)=9x(3x-9)^2 + (3x-9)^3\)