## ← Exponential Function Solution Solution - Differential Equations in Action

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Showing Revision 2 created 05/24/2016 by Udacity Robot.

1. If you plug in x equals zero, the result has to be five.
2. Look at this term. If x equals zero, we have e to the power of zero, which is one.
3. To turn that into five, this has to be five.
4. In the other box, we need to minus three. This is easy to see if you know about the chain rule.
5. The inner derivative provides us with a factor of minus three, which we need.
6. If you don't know the chain rule, you can argue as follows:
7. We start with e to the power of X. The standard exponential function.
8. Next, let's form e to the power of minus x, which means to flip that blue curve along the y axis.
9. To compute E to the minus X for a specific value of X, we can compute E to the X for minus that value of X.
10. So we're mirroring the blue curve to get the green curve.
11. This additional factor of three accelerates the green curve. E to the power of minus 3x is three times as fast,
12. if you will, as E to the power of minus X. To compute E to the power of minus 3x for this X,
13. we can compute E to the power of minus X for three times that X.
14. And now look at the derivatives. By going from the blue curve--the regular exponential function--
15. --to the green curve, we are changing the sign of the derivative.
16. Here the derivative is positive. Now it's negative all of the time.
17. And with that additional factor of three in the exponent, you're boosting the derivative by a factor of three.
18. Look at these triangles. The horizontal leg of this slope triangle is reduce by a factor of three,
19. meaning that the slope increases by a factor of three.