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Chapter 2: Q. 71 (page 223)

Prove that if f is an exponential function, then f'(x) is proportional to f(x).

Short Answer

Expert verified

It is proved thatf'(x) is proportional to f(x).

Step by step solution

01

Step 1. Given Information  

The given statement is if f is an exponential function, then f'(x) is proportional to f(x).

02

Step 2. Proof  

Let f(x) is proportional to exwhich can be written as f(x)=kex

Now, we will differentiate the function, we get,

f'(x)=ddxkexf'(x)=kexf'(x)=kf(x)f'(x)鈭�f(x)

Hence, Proved.

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Most popular questions from this chapter

Use the definition of the derivative to find f' for each function f in Exercises 39-54

f(x)=-x3-x+1

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

The tangent line to f(x)=x2 atx=0

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c)for each function f and value x=c in Exercises 23鈥�38.

23.f(x)=x2,x=-3

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

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)=(x4-8)(1-3x5);f(0)=2
See all solutions

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