PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Remember that the derivative of ex is itself, ex. So, by using the sum rule, you can calculate the derivative of a function that involves an exponential term. For example, let f(x)=7x3-8x2+2+4ex. By using the power rule, the derivative of 7x3 is 3*7x2=21x2, the derivative of -8x2 is 2*(-8)x=-16x, and the derivative of 2 is 0. Then, using what we know about the derivative of ex, we know that the derivative of 4ex is simply 4ex. So, the derivative of the entire function is f'(x)=21x2-16x+4ex.
Let’s try a couple of harder problems. We are asked to differentiate each function. First we have f(x) equals 2e to the x plus x². So f' would be the derivative and that’s going to be the derivative with respect to x of 2e to the x plus x².
Now I can separate these using the sum rule; the derivative of two e to the x plus the derivative of x². And then I can use the constant multiple rule to pull these two out. So I get 2 times the derivative of e to the x, plus the derivative of x² remember that it's 2x.
One of the big results of this section was the derivative of e to the x is just e to the x. So this is 2 times e to the x plus 2x. Let’s take a look at another one. G(x) equals 4 minus e to the 2 plus x. Now this one is tricky because it’s not clear exactly what to do with the e to the 2 plus x. It’s not e to the x. I need it to be e to the x. One thing I can do is rewrite this function as 4 minus. And then using properties of exponents, this would be e to the 2 times e to the x. E to the 2, e². This is just a constant, so I differentiate using constant multiple rule to pull out. So let’s do that.
The derivative g'(x), is going to be the derivative with respect to x of 4 minus e² times e to the x. I will separate this using sum rule. Get the derivative with respect to x of 4 plus the derivative with respect to x of –e², e to the x.
The derivative with respect of 4, you can think of as a linear function 0x plus 4. But its slope is going to be 0, plus and then I can pull this constant. Let me actually just write minus e². I'm pulling this constant outside times the derivative of e to the x. And so I get minus e², times e to the x. Again this is just e to the x.
And if you like you can put this back in the original form that it came in. We can multiply these together and get e to the 2 plus x. And so that’s our final answer. Minus e to the 2 plus x.