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Chapter 2: Q 12. (page 238)

Find the derivatives of the function:f(x)=3x2x+11-x.

Short Answer

Expert verified

The derivative of f(x)=3x2x+11-xis-3x2+9x+32x+1(1-x)2.

Step by step solution

01

Given information 

The given expression isf(x)=3x2x+11-x.

02

Simplification  

f'(x)=ddx3x2x+11-x=1-xddx3x2x+1-3x2x+1ddx{1-x}(1-x)2=(32x-1+3x脳122x+1脳2)-{1-x-3x2x+1脳-11-x2=-3x2+9x+32x+11-x2Hence,f'(x)=-3x2+9x+32x+11-x2.

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Most popular questions from this chapter

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x-4(2x+1)6

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the z鈫�xdefinition of the derivative

b) by using theh鈫�0definition of the derivative

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 Linda鈥檚 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 Linda鈥檚 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 Linda鈥檚 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 Linda鈥檚 (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Linda鈥檚 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 f'for each function fin Exercises 34-59

f(x)=1x3

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.
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