# How do you use the quotient rule to show that #1/f(x)# is decreasing given that f(x) is a positive increasing function defined for all x?

By signing up, you agree to our Terms of Service and Privacy Policy

To show that ( \frac{1}{f(x)} ) is decreasing given that ( f(x) ) is a positive increasing function defined for all ( x ), we can use the quotient rule and the fact that ( f(x) ) is increasing.

Using the quotient rule:

[ \frac{d}{dx} \left( \frac{1}{f(x)} \right) = \frac{-f'(x)}{(f(x))^2} ]

Since ( f(x) ) is increasing, ( f'(x) > 0 ) for all ( x ). Also, since ( f(x) ) is positive, ( (f(x))^2 > 0 ). Therefore, ( \frac{-f'(x)}{(f(x))^2} < 0 ).

Hence, ( \frac{d}{dx} \left( \frac{1}{f(x)} \right) < 0 ), which implies that ( \frac{1}{f(x)} ) is decreasing.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the derivative of #y=6sin(2t) + cos(4t)#?
- How can I find the derivative of the inverse of #f(x)= x^3+x+1# at x=11?
- What is the derivative of #cos(a^3+x^3)#?
- How do you find the derivative of the function #g(t)=1/sqrtt#?
- How do you find #dy/dx# by implicit differentiation given #y=cos(x+y)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7