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Chapter 2: Q. 1 (page 221)

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Short Answer

Expert verified

When fxis an exponential function of the form for some constant 'k' is given.

When b≠1 and all appropriate values of 'x' for any constant b>0is given.

Also if localid="1649875555882" fx,f-1x are inverse functions and differentiable, then for all appropriate values of 'x' is given.

Step by step solution

01

Step 1. When fx is an exponential function of the form.

Consider the given question,

f'x=kfx for some constant 'k' if and only if fxis an exponential function of the form.

ddxex=exddxbx=bx.lnbddxekx=k.ekx

02

Step 2. When b≠1 and all appropriate values of 'x'.

For any constant 'k' any constant b>0with b≠1 and all appropriate values of 'x'.

ddxlogbx=1logbxddxlnx=1xddxlnx=1x

If localid="1649875446650" fx,f-1x are inverse functions and differentiable, then for all appropriate values of 'x',

f-1'x=1f'f-1x

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Most popular questions from this chapter

State the chain rule for differentiating a composition g(h(x))of two functions expressed

(a) in “prime” notation and

(b) in Leibniz notation.

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

a) by using the z→xdefinition of the derivative

b) by using theh→0definition of the derivative

Suppose u(x)=3x2+1and f(u)=u2+3u51-u. Use the chain rule to find role="math" localid="1648356625815" ddx(f(u(x))) without first finding the formula for f(u(x)).

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

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

24.f(x)=x3,x=1

Prove that if f is any cubic polynomial function f(x)=ax3+bx2+cx+dthen the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows

a=f"'(0)6;b=f"(0)2;c=f'(0);d=f(0)
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