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Chapter 3: Problem 28

Find the derivative of the following functions. $$y=3 x^{3} \ln x-x^{3}$$

Short Answer

Expert verified
In the given function $$y = 3x^3\ln x - x^3$$, we found the derivative using the product rule for the first term and the power rule for the second term. After differentiating and simplifying, we obtained $$y' = 9x^2\ln x$$.

Step by step solution

01

Identify terms and rules to apply

Identify the terms and the differentiation rules applicable. In this case, we have two terms in the function: $$3 x^{3} \ln x$$ and $$-x^{3}$$. We will use the product rule for the first term and the power rule for the second term.
02

Apply the product rule on the first term

To differentiate the first term $$3 x^{3} \ln x$$, apply the product rule: $$(u v)'=u' v + u v'$$, where $$u=3 x^{3}$$ and $$v=\ln x$$. First, find $$u'$$: $$u'= \frac{d(3x^{3})}{dx} = 9x^2$$ Now, find $$v'$$: $$v' = \frac{d(\ln x)}{dx} = \frac{1}{x}$$ Apply the product rule: $$(3 x^{3} \ln x)' = (9x^2)(\ln x) + (3x^3)(\frac{1}{x})$$
03

Apply the power rule on the second term

To differentiate the second term $$-x^3$$, apply the power rule: $$\frac{d(-x^{3})}{dx} = -3x^2$$
04

Combine the results

Combine the results from Step 2 and Step 3 to find the derivative of the whole function: $$y' = (3 x^{3} \ln x)' - x^{3}' = (9x^2)(\ln x) + (3x^3)(\frac{1}{x}) -3x^2$$
05

Simplify the derivative

Simplify the derivative by combining like terms: $$y' = 9x^2\ln x + 3x^2 - 3x^2 = 9x^2\ln x$$ So, the derivative of the given function $$y$$ with respect to $$x$$ is $$y' = 9x^2\ln x$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule in Differentiation
When you come across a function that is the product of two simpler functions, you'll often need to use the product rule to find its derivative. The product rule is a crucial tool in calculus. It tells us how to differentiate expressions where two functions are multiplied together. The formula for the product rule is:
  • If you have a function given by \[ y = u imes v \], where both \( u \) and \( v \) are functions of \( x \),
  • then the derivative of \( y \) with respect to \( x \) is: \[ y' = u'v + uv' \].
Here's a simple way to think about it:- Differentiate the first function, \( u \), and multiply it by the second function, \( v \). - Then add the product of the first function and the derivative of the second one, \( v' \).In practice, follow these steps step-by-step for clear and accurate results.
Power Rule for Derivatives
The power rule is one of the simplest yet most useful rules for differentiation. It's especially handy when dealing with polynomial functions. The power rule states:
  • For any function \( y = x^n \), where \( n \) is a constant number, the derivative of \( y \) with respect to \( x \) is: \[ \frac{d}{dx}(x^n) = nx^{n-1} \].
This rule is a quick shortcut for finding derivatives, allowing us to easily handle terms with variables raised to a power. Be mindful of any constant coefficients in front of polynomial terms, such as the \(-x^3\) in our original function. The power rule is applied by:- Bringing down the exponent as a coefficient.- Reducing the exponent by one. This process simplifies taking derivatives manually, making calculations faster and less error-prone.
Applications of Differentiation
Differentiation isn't just a purely abstract mathematical process. It's a powerful tool that plays a significant role in various fields such as physics, engineering, and economics. By finding the derivative of a function, you can determine:
  • Instantaneous rate of change: This is particularly useful in studying how changes occur rapidly in physics or economics, such as velocity, acceleration, or cost adjustments.
  • Critical points: By setting a derivative to zero, one can find where a function reaches its maximum or minimum values, aiding in optimization problems.
  • Graph shapes: Knowing how graphs behave through increasing or decreasing intervals can help in analyzing the behavior of functions.
The derivative of a function provides insightful information about the function itself and helps solve practical real-world problems, making it an indispensable tool across various disciplines.

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Most popular questions from this chapter

Use implicit differentiation to find\(\frac{d y}{d x}.\) $$y=x e^{y}$$

Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. $$\sin y+5 x=y^{2} ;(0,0)$$ (Graph cant copy)

Find \(d^{2} y / d x^{2}.\) $$x+y^{2}=1$$

Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of \(x.\) b. Graph the tangent lines on the given graph. \(4 x^{3}=y^{2}(4-x) ; x=2\) (cissoid of Diocles) (Graph cant copy)

Use implicit differentiation to find\(\frac{d y}{d x}.\) $$x+2 y=\sqrt{y}$$
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