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Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra.

2cosx3.

Short Answer

Expert verified

Thevalueofthederivativeis:-6x2sin(x3).

Step by step solution

01

Step 1. Given Information.

f(x)=2cosx3.

02

Step 2. General formulas for finding derivatives.

ddxxn=nxn-1,ddxsinx=cosx,ddxcosx=-sinx,ddxtanx=sec2x,ddx(secx)=secxtanx,ddx(cscx)=-cscxcotx,ddx(cotx)=-csc2x,Productrule:ddx(uv)=udvdx+vdudx,Quotientrule:ddxuv=vdudx-udvdxv2,Chainrule:ddxfog(x)=ddxf(g(x))×ddx(g(x).

03

Step 3. Solving derivative using above formulas.

f(x)=2cos(x3),Differentiatingusingthechainruleweget,ddx(f(x))=-2sin(x3)ddx(x3)=-2sin(x3)(3x2)=-6x2sin(x3).

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Most popular questions from this chapter

Taking the limit: We have seen that if f is a smooth function, then f'(c)f(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limh0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.

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

Last night Phil went jogging along Main Street. His distance from the post office t minutes after 6:00p.m. is shown in the preceding graph at the right.

(a) Give a narrative (that matches the graph) of what Phil did on his jog.

(b) Sketch a graph that represents Phil’s instantaneous velocity t minutes after 6:00p.m. Make sure you label the tick marks on the vertical axis as accurately as you can.

(c) When was Phil jogging the fastest? The slowest? When was he the farthest away from the post office? The closest to the post office?

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

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

Use theh0definition of the derivative to prove the power rule holds for positive integers powers

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