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Chapter 4: Problem 1

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. $$x^{4}-3 x^{3}+(1 / 2) x^{2}+7 x-\pi$$

Short Answer

Expert verified
\(f'(x) = 4x^{3} - 9x^{2} + x + 7\)

Step by step solution

01

Identify the Function

The function we need to differentiate is \(f(x) = x^{4} - 3x^{3} + \frac{1}{2}x^{2} + 7x - \pi\). This is a polynomial function where each term is a power of \(x\).
02

Apply the Power Rule

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). We will apply this rule to each term of the polynomial. For constants, the derivative is 0.
03

Differentiate Each Term

\(f(x) = x^{4} - 3x^{3} + \frac{1}{2}x^{2} + 7x - \pi\). Differentiate as follows:- \(x^{4}\) becomes \(4x^{3}\)- \(-3x^{3}\) becomes \(-9x^{2}\)- \(\frac{1}{2}x^{2}\) becomes \(x\)- \(7x\) becomes \(7\)- \( -\pi\) becomes \(0\) since the derivative of a constant is 0.
04

Write the Resultant Derivative

Combine the derivative of each term to form the derivative of the function:\(f'(x) = 4x^{3} - 9x^{2} + x + 7\).

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Key Concepts

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

Power Rule in Differentiation
The Power Rule is a fundamental tool in calculus for finding derivatives of functions that involve powers of a variable. Given a function of the form \( x^n \), where \( n \) is a real number, the Power Rule states that its derivative is \( nx^{n-1} \). This means that to find the derivative, you simply multiply the power by the coefficient and reduce the power by one.

For example:
  • If you have \( f(x) = x^4 \), the derivative \( f'(x) \) is \( 4x^3 \).
  • If you have \( g(x) = -3x^3 \), using the Power Rule, the derivative \( g'(x) \) is \( -9x^2 \).
Applying the Power Rule is especially effective for polynomial functions, as it allows you to find derivatives quickly and accurately.
Understanding Polynomial Functions
Polynomial functions are algebraic expressions that consist of variables raised to whole number exponents and coefficients. They take the general form \( a_n x^n + a_{n-1} x^{n-1} + \, \ldots \, + a_1 x + a_0 \), where each \( a_i \) is a constant and the highest exponent \( n \) is the degree of the polynomial.

The function given, \( f(x) = x^4 - 3x^3 + \frac{1}{2}x^2 + 7x - \pi \), is a polynomial of degree 4, as the highest power of \( x \) is 4. The terms include:
  • \( x^4 \), representing a quartic term.
  • \(-3x^3 \), representing a cubic term.
  • \(\frac{1}{2}x^2 \), representing a quadratic term.
  • \(7x \), representing a linear term.
  • \(-\pi \), a constant term.
Knowing the structure of polynomial functions is crucial for applying the Power Rule effectively to each term.
Basics of Derivatives
In calculus, a derivative represents the rate at which a function is changing at any given point. It is a fundamental concept that helps us understand how functions behave by giving insights into their slopes and rates of change.

In the context of our polynomial function, what we are doing is calculating the derivative, \( f'(x) \). This involves applying rules like the Power Rule to find the derivatives of each term, such as:
  • \(4x^3\) is the derivative of \(x^4\).
  • \(-9x^2\) is what you get from \(-3x^3\).
  • \(x\) is the derivative of \(\frac{1}{2}x^2\).
  • \(7\) is the derivative of \(7x\).
  • The derivative of a constant like \(-\pi\) is \(0\).
This makes the resultant derivative function \( f'(x) = 4x^3 - 9x^2 + x + 7 \). Derivatives are key to solving many problems in science and engineering, as they can describe motion, growth rates, and other dynamic phenomena.

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Most popular questions from this chapter

Compute the derivative of \(\frac{x^{2}+5 x-3}{x^{5}-6 x^{3}+3 x^{2}-7 x+1} .\)

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. $$\frac{\sqrt{25-x^{2}}}{x}$$

Differentiate the following functions. (a) \(y=e^{3 x}+e^{-x}+e^{2}\) (b) \(y=e^{2 x} \cos 3 x\) (c) \(f(x)=\tan \left(x+e^{x}\right)\) (d) \(g(x)=\frac{e^{x}}{e^{x}+2}\) (e) \(y=\ln (2+\sin x)-\sin (2+\ln x)\) (f) \(f(x)=e^{x^{\pi}}+x^{\pi^{e}}+\pi^{e^{x}}\) (g) \(y=\log _{a}\left(b^{x}\right)+b^{\log _{a} x}\), where a and \(b\) are positive real numbers and \(a \neq 1\). (h) \(y=\left(x^{2}+1\right)^{x^{3}+1}\) (i) \(y=\left(x^{2}+e^{x}\right)^{1 / \ln x}\) (j) \(y=\frac{x \sqrt{x^{2}+x+1}}{(2+\sin x)^{4}(3 x+5)^{7}}\)

Two curves are orthogonal if at each point of intersection, the angle between their tangent lines is \(\pi / 2 .\) Two families of curves, \(\mathscr{A}\) and \(\mathscr{B}\), are orthogonal trajectories of each other if given any curve \(\mathrm{C}\) in \(\mathscr{A}\) and any curve \(D\) in \(\mathscr{B}\) the curves \(\mathrm{C}\) and \(D\) are orthogonal. For example, the family of horizontal lines in the plane is orthogonal to the family of vertical lines in the plane. (a) Show that \(x^{2}-y^{2}=5\) is orthogonal to \(4 x^{2}+9 y^{2}=72 .\) (Hint: You need to find the intersection points of the two curves and then show that the product of the derivatives at each intersection point is \(-1 .)\) (b) Show that \(x^{2}+y^{2}=r^{2}\) is orthogonal to \(y=m x .\) Conclude that the family of circles centered at the origin is an orthogonal trajectory of the family of lines that pass through the origin. Note that there is a technical issue when \(m=0 .\) The circles fail to be differentiable when they cross the x-axis. However, the circles are orthogonal to the x-axis. Explain why. Likewise, the vertical line through the origin requires a separate argument. (c) For \(k \neq 0\) and \(c \neq 0\) show that \(y^{2}-x^{2}=k\) is orthogonal to \(y x=c .\) In the case where \(k\) and \(c\) are both zero, the curves intersect at the origin. Are the curves \(y^{2}-x^{2}=0\) and \(y x=0\) orthogonal to each other? (d) Suppose that \(m \neq 0 .\) Show that the family of curves \(\\{y=m x+b \mid b \in \mathbb{R}\\}\) is orthogonal to the family of curves \(\\{y=-(x / m)+c \mid c \in \mathbb{R}\\}\).

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. $$(4-x)^{3}$$
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