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Chapter 3: Problem 34

Find the derivatives of the function. $$y=x^{-3 / 3}+\pi^{3 / 2}$$

Short Answer

Expert verified
The derivative is \(-x^{-2}\).

Step by step solution

01

Simplify the Exponent

First, simplify the exponent in the term \( x^{-3/3} \). This simplifies to \( x^{-1} \) because the fraction \( -3/3 \) simplifies to -1.
02

Use the Power Rule for Differentiation

The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Apply this rule to find the derivative of \( x^{-1} \). Differentiating, we get \( -1 \cdot x^{-1-1} = -x^{-2} \).
03

Derive the Second Term

Identify that \( \pi^{3/2} \) is a constant term. The derivative of a constant is 0.
04

Combine the Derivatives

Combine the derivatives you found in the previous steps. The derivative of the function \( y = x^{-3/3}+\pi^{3/2} \) is \( -x^{-2} + 0 = -x^{-2} \).

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Key Concepts

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

Power Rule
The power rule is a fundamental principle in calculus, used to find the derivative of a function of the form \( x^n \). It simplifies the process of differentiation by establishing that the derivative of \( x^n \) is \( nx^{n-1} \). This rule is particularly useful because it streamlines the differentiation of polynomials.
  • When applying the power rule, it's important to correctly identify the exponent \( n \).
  • If the exponent is negative or a fraction, as in the expression \( x^{-3/3} \), it is advisable to simplify it first. For example, \( x^{-3/3} \) simplifies to \( x^{-1} \).
In practice, recognizing when to use the power rule involves:
  • Identifying terms that can be expressed in the form of \( x^n \).
  • Applying the rule iteratively to each applicable term in a polynomial or sum.
The power rule doesn't apply directly to all functions, but it's a key tool for polynomial expressions.
Derivative
In calculus, a derivative represents the rate at which a function is changing at any given point. It acts as a fundamental building block in understanding how functions behave. Differentiation, the process of finding a derivative, enables you to determine the slope of the tangent line to a graph at a particular point.
  • For a function \( y = f(x) \), the derivative \( y' = f'(x) \) provides a mathematical way to express the rate of change.
  • Not all functions change uniformly, hence calculative approaches like the power rule help ascertain these rates.
The derivative has several applications:
  • It helps find velocity and acceleration in physics, as these are derivatives of position with respect to time.
  • In economics, derivatives assist in calculating marginal costs and optimizing resources.
  • They are pivotal in determining maxima and minima of functions, essential in various real-world optimizations.
Being adept at finding derivatives involves understanding and applying rules like the power rule and recognizing constant functions for ease of computation.
Constant Function
A constant function is one that remains the same regardless of the input value. Mathematically, a constant function is expressed as \( f(x) = c \), where \( c \) is a constant.
  • The distinguishing feature of constant functions is their derivative, which is always zero.
  • This happens because there is no change in the value of the function over its domain, hence the slope of the graph is zero.
In the problem given, \( \pi^{3/2} \) is identified as a constant.
  • This term doesn鈥檛 change with \( x \), making it straightforward to differentiate.
  • Applying this, the derivative of \( \pi^{3/2} \) is \( 0 \).
Constant functions simplify analysis in calculus by providing intuitive insights into equations, where straightforward terms do not affect the rate of change in a function. Understanding constant functions is important for quickly identifying parts of an expression that contribute zero to the derivative.

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Most popular questions from this chapter

Graph the function \(y=2 \cos 2 x\) for \(-2 \leq x \leq 3.5 .\) Then, on the same screen, graph $$y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$ for \(h=1.0,0.5,\) and \(0.2 .\) Experiment with other values of \(h\) including negative values. What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\left(x^{2}+1\right)(x-1)^{2}}$$

Use the identity $$\csc ^{-1} u=\frac{\pi}{2}-\sec ^{-1} u$$ to derive the formula for the derivative of \(\csc ^{-1} u\) in Table 3.1 from the formula for the derivative of \(\sec ^{-1} u\)

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

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=\sin x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \quad x_{0}=1$$
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