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Chapter 7: Problem 16

Find the derivative of the function. $$ y=2 x^{3}-x^{2}+3 x-1 $$

Short Answer

Expert verified
\[ y' = 6x^2 - 2x + 3 \]

Step by step solution

01

Apply the Power Rule to the First Term

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). In the first term, the power of \(x\) is 3. So, applying the power rule gives the derivative as \(3 * 2x^{3-1} = 6x^2\).
02

Apply the Power Rule to the Second Term

In the second term, the power of \(x\) is 2. Applying the power rule gives the derivative as \(2 * 1x^{2-1} = 2x\). As this term is subtracted in the original function, it will also be subtracted in the derivative.
03

Apply the Power Rule to the Third Term

In the third term, the power of \(x\) is 1. Applying the power rule gives the derivative as \(1 * 3x^{1-1} = 3\).
04

Derive the Constant

The derivative of a constant term is simply 0. So, the derivative of -1 is 0.
05

Combine the Three Terms

Add the derived terms together to form the derivative of the complete function. Subtract the second term because it was subtracted in the original function. The derivative is \(6x^2 - 2x + 3\).

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Most popular questions from this chapter

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{3}, u=3 x^{2}-2 $$

You are managing a store and have been adjusting the price of an item. You have found that you make a profit of \(\$ 50\) when 10 units are sold, \(\$ 60\) when 12 units are sold, and \(\$ 65\) when 14 units are sold. (a) Fit these data to the model \(P=a x^{2}+b x+c\). (b) Use a graphing utility to graph \(P\). (c) Find the point on the graph at which the marginal profit is zero. Interpret this point in the context of the problem.

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. $$ f(x)=\sqrt{x^{2}-2 x+1} $$

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. $$ f(x)=2\left(x^{2}-1\right)^{3} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{2 x}{x+1}} $$
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