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Chapter 4: Problem 4

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=-3 x^{4}+6 x^{2}-2 $$

Short Answer

Expert verified
The derivative is \(-12x^3 + 12x\).

Step by step solution

01

Identify the Function and Power Rule

We are given the function \( f(x) = -3x^4 + 6x^2 - 2 \). The task is to differentiate this function with respect to \( x \). We will use the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
02

Differentiate Each Term

Apply the power rule term by term: 1. For \(-3x^4\), the derivative is \( -3 imes 4x^{4-1} = -12x^3 \).2. For \(6x^2\), the derivative is \( 6 imes 2x^{2-1} = 12x \).3. The derivative of a constant, \(-2\), is 0.
03

Combine the Derivatives

Combine the derivatives of each term to find the derivative of the entire function:- The derivative of \(-3x^4\) is \(-12x^3\).- The derivative of \(6x^2\) is \(12x\).- The derivative of \(-2\) is 0.Thus, the derivative \( f'(x) \) is \(-12x^3 + 12x \).

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Key Concepts

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

Understanding the Power Rule in Differentiation
The power rule is a key concept in calculus that makes finding the derivative of polynomial terms straightforward. When you have a function in the form of \( f(x) = ax^n \), the power rule helps you easily determine its derivative. By applying this rule, the derivative is given by \( f'(x) = anx^{n-1} \). Here's how it works:
  • Identify the exponent (n): Look at the power of the variable \( x \) in each term of your function.
  • Multiply by the exponent: Multiply the entire term by this exponent.
  • Decrease the exponent by one: Reduce the exponent by one to find the new exponent for \( x \).
The power rule is particularly useful because it's quick and works directly on each term, making it ideal for differentiating polynomial functions.
What is a Polynomial Function?
Polynomial functions are algebraic expressions that involve sums of powers of a variable. These can be simple or complex, containing various terms with coefficients and exponents. A typical polynomial function looks like \( f(x) = ax^n + bx^{n-1} + \, \ldots \, + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( n \) is a non-negative integer.
  • Degrees: The degree of a polynomial is the highest power of \( x \) present in the function. It tells you how many times polynomials can be multiplied to form the polynomial.
  • Terms: Each part of the polynomial separated by a plus or minus sign is a term. For instance, \(-3x^4\), \(6x^2\), and \(-2\) are terms of the given polynomial.
  • Coefficients: The numbers multiplying the powers of \( x \) are known as coefficients, such as -3, 6, and -2 in the example function.
Differentiating polynomial functions is often straightforward, thanks to the power rule, which can be applied term by term.
Steps in Derivative Calculation
Calculating the derivative of a polynomial function involves a systematic approach to differentiating each term. Let's break down how to compute the derivative of the example function \( f(x) = -3x^4 + 6x^2 - 2 \):
  • Apply the power rule to each term: Take \(-3x^4\), for instance. Use the power rule: multiply -3 by 4 (the exponent of \( x \)), and reduce the exponent from 4 to 3. This gives \(-12x^3\).
  • Repeat for other terms: For \(6x^2\), multiply 6 by 2, and then reduce the exponent of \( x \) by one, resulting in \(12x\).
  • Derive constants directly: A constant like -2 has a derivative of 0 because constants do not change as \( x \) changes.
  • Combine all derivatives: Put the results together to form the complete derivative of the polynomial function, which is \(-12x^3 + 12x\).
Using this method ensures accuracy and simplicity in obtaining the derivative of any polynomial.

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Most popular questions from this chapter

Suppose that the rate of change of the size of a population is given by $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right) $$ where \(N=N(t)\) denotes the size of the population at time \(t\) and \(r\) and \(K\) are positive constants. Find the equilibrium size of the population-that is, the size at which the rate of change is equal to \(0 .\) Use your answer to explain why \(K\) is called the carrying capacity.

Compute \(f(c+h)-f(c)\) at the indicated point. $$ f(x)=\sqrt{x} ; c=4 $$

Use the formal definition of the derivative to find the derivative of \(y=5 x^{2}\) at \(x=-1\). (b) Show that the point \((-1,5)\) is on the graph of \(y=5 x^{2}\), and find the equation of the tangent line at the point \((-1,5)\). (c) Graph \(y=5 x^{2}\) and the tangent line at the point \((-1,5)\) in the same coordinate system.

Suppose that the per capita growth rate of a population is \(2 \% ;\) that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.02$$ Suppose also that the population size at time \(t=2\) is equal to 50. Use a linear approximation to compute the population size at time \(t=2.1\).

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} x & \text { for } x \leq 0 \\ x+1 & \text { for } x>0 \end{array}\right. $$
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