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Use the differentiation rules developed in this section to find

the derivatives of the functions

f(x)=4-x7

Short Answer

Expert verified

The derivative off(x)=4-x7is-7x6

Step by step solution

01

Step1. Given information

Here function f(x)=4-x7is given.

we have to find out the derivatives of this function

02

Step2. finding the derivatives of the function

By using the power rule, we can solve this problem.

Power rule: For any nonzero rational number k,

ddxxk=kxk-1

Thus using this power rule when we do the differentiation , we get

ddx(4-x7)=d(4)dx-d(x7)dx=0-7x6sincederivativeofaconstantis0=-7x6thusthederivativeofgivenfunctionis-7x6

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Most popular questions from this chapter

Write down a rule for differentiating a composition f(u(v(w(x))))of four functions

(a) in โ€œprimeโ€ notation and

(b) in Leibniz notation.

Use the limit you just found to calculate the exact slope of the tangent line to f(x)=x2at x=4. Obviously you should get the same final answer as you did earlier.

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?

The following reciprocal rules tells us hoe to differentiate the reciprocal of a function

ddx(1f(x))=-1[f(x)]2

Prove this using

a) definition of the derivative

b) by using the quotient rule

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=x(4-2x);f(0)=0

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