Chapter 3

Give an example in which one dimension of a geometric figure changes and produces a corresponding change in the area or volume of the figure.

Use \(x=e^{y}\) to explain why \(\frac{d}{d x}(\ln x)=\frac{1}{x},\) for \(x>0\).

$$\text { State the derivative formulas for } \sin ^{-1} x, \tan ^{-1} x, \text { and } \sec ^{-1} x$$

Two equivalent forms of the Chain Rule for calculating the derivative of \(y=f(g(x))\) are presented in this section. State both forms.

How do you find the derivative of the product of two functions that are differentiable at a point?

Explain why \(f^{\prime}(x)\) could be positive or negative at a point where \(f(x) > 0\).

Explain how implicit differentiation can simplify the work in a related-rates problem.

Complete the following statement. If \(\frac{d y}{d x}\) is large, then small changes in \(x\) result in relatively __________ changes in the value of \(y\).

Sketch the graph of \(f(x)=\ln |x|\) and explain how the graph shows that \(f^{\prime}(x)=\frac{1}{x}\).

What is the slope of the line tangent to the graph of \(y=\sin ^{-1} x\) at \(x=0 ?\)

Explain the differences between computing the derivatives of functions that are defined implicitly and explicitly.

How do you find the derivative of the quotient of two functions that are differentiable at a point?

Assume the derivatives of \(f\) and \(g\) exist. In this section, we showed that the rule \(\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\) is valid for what values of \(n ?\)

Explain why \(f(x)\) could be positive or negative at a point where \(f^{\prime}(x )< 0\).

Explain why the slope of a secant line can be interpreted as an average rate of change.

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

Complete the following statement: If \(\frac{d y}{d x}\) is small, then small changes in \(x\) result in relatively _________ changes in the value of \(y\).

Show that \(\frac{d}{d x}(\ln k x)=\frac{d}{d x}(\ln x),\) where \(x>0\) and \(k\) is a positive real number.

What is the slope of the line tangent to the graph of \(y=\tan ^{-1} x\) at \(x=-2 ?\)

Why are both the \(x\) -coordinate and the \(y\) -coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function?

Fill in the blanks. The derivative of \(f(g(x))\) equals \(f^{\prime}\) evaluated at ________ multiplied by \(g^{\prime}\) evaluated at ________

State the Extended Power Rule for differentiating \(x^{n}\). For what values of \(n\) does the rule apply?

Explain why the Quotient Rule is used to determine the derivative of \(\tan x\) and \(\cot x\)

Assume the derivatives of \(f\) and \(g\) exist. Give a nonzero function that is its own derivative.

If \(f\) is differentiable at \(a,\) must \(f\) be continuous at \(a ?\)

Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.

What is the difference between the velocity and speed of an object moving in a straight line?

State the derivative rule for the exponential function \(f(x)=b^{x}\) How does it differ from the derivative formula for \(e^{x} ?\).

How are the derivatives of \(\sin ^{-1} x\) and \(\cos ^{-1} x\) related?

Show two ways to differentiate \(f(x)=1 / x^{10}\)

How can you use the derivatives \(\frac{d}{d x}(\sin x)=\cos x\) \(\frac{d}{d x}(\tan x)=\sec ^{2} x,\) and \(\frac{d}{d x}(\sec x)=\sec x \tan x\) to remember the derivatives of \(\cos x, \cot x,\) and \(\csc x ?\)

Assume the derivatives of \(f\) and \(g\) exist. How do you find the derivative of the sum of two functions \(f+g ?\)

For a given function \(f,\) what does \(f^{\prime}\) represent?

Expanding square The sides of a square increase in length at a rate of \(2 \mathrm{m} / \mathrm{s}\). a. At what rate is the area of the square changing when the sides are \(10 \mathrm{m}\) long? b. At what rate is the area of the square changing when the sides are 20 m long? c. Draw a graph that shows how the rate of change of the area varies with the side length.

Define the acceleration of an object moving in a straight line.

State the derivative rule for the logarithmic function \(f(x)=\log _{b} x .\) How does it differ from the derivative formula for \(\ln x ?\)

Suppose \(f\) is a one-to-one function with \(f(2)=8\) and \(f^{\prime}(2)=4 .\) What is the value of \(\left(f^{-1}\right)^{\prime}(8) ?\)

Carry out the following steps. a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find the slope of the curve at the given point. $$x^{4}+y^{4}=2 ;(1,-1)$$

Identify the inner and outer functions in the composition \(\left(x^{2}+10\right)^{-5}\).

What is the derivative of \(y=e^{k x} ?\) For what values of \(k\) does this rule apply?

Let \(f(x)=\sin x .\) What is the value of \(f^{\prime}(\pi) ?\)

Assume the derivatives of \(f\) and \(g\) exist. How do you find the derivative of a constant multiplied by a function?

Given a function \(f\) and a point \(a\) in its domain, what does \(f^{\prime}(a)\) represent?

Shrinking square The sides of a square decrease in length at a rate of \(1 \mathrm{m} / \mathrm{s}\) a. At what rate is the area of the square changing when the sides are \(5 \mathrm{m}\) long? b. At what rate are the lengths of the diagonals of the square changing?

An object moving along a line has a constant negative acceleration. Describe the velocity of the object.

Explain how to find \(\left(f^{-1}\right)^{\prime}\left(y_{0}\right),\) given that \(y_{0}=f\left(x_{0}\right)\)

Explain why \(b^{x}=e^{x \ln b}\).

Carry out the following steps. a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find the slope of the curve at the given point. $$x=e^{y} ;(2, \ln 2)$$

Express \(Q(x)=\cos ^{4}\left(x^{2}+1\right)\) as the composition of three functions; that is, identify \(f, g,\) and \(h\) so that \(Q(x)=f(g(h(x))).\)

Show two ways to differentiate \(f(x)=(x-3)\left(x^{2}+4\right)\)