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Chapter 7: Problem 20

Find the derivative of the function. $$ g(x)=4 \sqrt[3]{x}+2 $$

Short Answer

Expert verified
The derivative of the function \(g(x) = 4\sqrt[3]{x} + 2\) is \(g'(x) = 4/3*x^{-2/3}\).

Step by step solution

01

Identify the form of the function

Firstly, recognize that the function can be rewritten as a sum of simpler functions. Here, \(g(x) = 4x^{1/3} + 2\). The function \(f(x) = x^n\) is a basic function with a known formula for the derivative.
02

Apply the power rule

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Thus, using the power rule, the derivative of \(4x^{1/3}\) is \(4*1/3*x^{1/3-1}\) which simplifies to \(4/3*x^{-2/3}\).
03

Take the derivative of the constant function

The derivative of a constant function is zero. Thus, the derivative of constant 2 is 0.
04

Combine the results

Since our function is a sum of two simpler functions, its derivative is the sum of the derivatives of the simpler functions. Therefore, the derivative of our original function \(g(x)\) is \(4/3*x^{-2/3} + 0\) or simply, \(4/3*x^{-2/3}\).

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Most popular questions from this chapter

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{2}{1-x^{3}} $$

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. $$ f(x)=\left(4-3 x^{2}\right)^{-2 / 3} $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=(4-3 x)^{-5 / 2} $$

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=2 \sqrt{u}, u=5 x+9 $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=(3 x+1)^{-1} $$
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