A Fun Little Party Trick

Calculus Level 2

d d x ( A x ) = A x \dfrac{d}{dx} (A^{x}) = A^{x}

Suppose that for a particular value of the constant A A the equation above is true for all x x . Determine the value of A A to three decimal places.

Note: You probably know right away what the answer is. But try to derive it using difference quotients, as suggested above, and then post your solution. Good for impressing friends who haven't seen it before.


The answer is 2.718.

Steven Chase by Steven Chase , 37, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Steven Chase
Aug 16, 2016

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Md Zuhair
Aug 16, 2016

Here we get d d x ( A x ) = A x \frac{d}{dx} (A^{x}) = A^{x} , Then d ( A x ) = A x . d x d(A^x) = A^x . dx Then Integrating both Sides we get A x = A x / l o g A A^x = A^x / log A

Then l o g A = 1 = A = e = 2.71828182846 log A =1 = A= e = 2.71828182846

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Matt Mcc
Aug 19, 2016

d d x \frac{d}{dx} ( A x A^{x} ) = A x A^{x}

First differentiate by using the exponential rule of differentiation.

ln( A x A^{x} ) A x A^{x} = A x A^{x}

Simplify both sides.

ln( A x A^{x} ) = 1

Re-arrange using log laws.

e 1 e^{1} = A

Solve.

A = e ≈ 2.718 \boxed{A = e ≈ 2.718}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I think its lnA instead of lnA^x

Prince Loomba - 4 years, 9 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...