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Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

f(x)=(3xx)-2

Short Answer

Expert verified

The derivative of the given function is-13x4.

Step by step solution

01

Step 1. Given Information

The given expression isf(x)=(3xx)-2.

02

Step 2. Simplify the function

Simplify the function.

f(x)=3x32-2=3-2x-3=x-39

03

Step 2. Find the derivative

  • Apply the constant multiple rule of derivative,(kf)'(x)=kf'(x).

f'(x)=ddx(x-39)=19ddxx-3

  • Apply the power rule of derivative, (xn)'=nxn-1.

f'(x)=19(-3x-4)=-x-43=-13x4

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Most popular questions from this chapter

Use (a) the hโ†’0definition of the derivative and then

(b) the zโ†’cdefinition of the derivative to find f'(c)for each function f and value x=c in Exercises 23โ€“38.

27.f(x)=1-x3,x=-1

Think about what you did today and how far north you were from your house or dorm throughout the day. Sketch a graph that represents your distance north from your house or dorm over the course of the day, and explain how the graph reflects what you did today. Then sketch a graph of your velocity.

Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park tminutes after she begins her jog is given by the function s(t)shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Lindaโ€™s distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Lindaโ€™s distance from the oak tree the greatest: the first 10minutes, the second 10minutes, or the last10minutes?

(c) Use the graph of s(t)to estimate Lindaโ€™s average velocity during the 5-minute interval fromt=5tot=10. What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Lindaโ€™s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Lindaโ€™s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.

The line that passes through the point (3,2)and is parallel to the tangent line to f(x)=1x at x=-1.

Use (a) the hโ†’0definition of the derivative and then

(b) the zโ†’cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23โ€“38.

25.f(x)=1x,x=-1

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