Home > Calculus > Quotient Rule

What is the derivative of # 9e^x/x#?

Answer 1

Luke Alvarez

Use the Quotient Rule ...

#f'(x)=9[(xe^x-e^x)/x^2]=(9e^x)/x^2(x-1)#

hope that helped

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

Answer 2

William Abdalla

To find the derivative of ( \frac{9e^x}{x} ), you can use the quotient rule. The quotient rule states that for functions ( u(x) ) and ( v(x) ), the derivative of ( \frac{u(x)}{v(x)} ) is given by:

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

Where ( u'(x) ) and ( v'(x) ) denote the derivatives of ( u(x) ) and ( v(x) ) with respect to ( x ) respectively.

In this case, ( u(x) = 9e^x ) and ( v(x) = x ). Thus, ( u'(x) = 9e^x ) and ( v'(x) = 1 ).

Now, applying the quotient rule:

[ \frac{d}{dx}\left(\frac{9e^x}{x}\right) = \frac{(9e^x)(x) - (9e^x)(1)}{(x)^2} ]

[ = \frac{9xe^x - 9e^x}{x^2} ]

[ = \frac{9e^x(x - 1)}{x^2} ]

So, the derivative of ( \frac{9e^x}{x} ) is ( \frac{9e^x(x - 1)}{x^2} ).

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

Answer from HIX Tutor

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.