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Chapter 7: Problem 59

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

Short Answer

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

Step by step solution

01

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 \(v(x)=(3x-9)^3\)
02

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 \(v'(x)=9(3x-9)^2\).
03

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 \(f'(x)=x*9(3x-9)^2 + (3x-9)^3*1\)
04

Simplify the Expression

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

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Most popular questions from this chapter

The cost of producing \(x\) units of a product is given by \(C=x^{3}-15 x^{2}+87 x-73, \quad 4 \leq x \leq 9\) (a) Use a graphing utility to graph the marginal cost function and the average cost function, \(C / x\), in the same viewing window. (b) Find the point of intersection of the graphs of \(d C / d x\) and \(C / x\). Does this point have any significance?

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=3(9 x-4)^{4} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{6-5 x}{x^{2}-1}\right)^{2} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\left(x^{5}-3 x\right)\left(\frac{1}{x^{2}}\right) $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\sqrt{5 x-2} $$
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