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Derivatives of Exponential Functions - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
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 do a problem that involves the derivatives of exponential functions. Now let’s recall that the derivative formula is the derivative with respect to x of a to the x, is natural log of a times a to the x. So here is a simple example f of x equals 10 to the x. What will the derivative of that be?

F'(x) would be the derivative of 10 to the x. So that’s our a value. And according to this, the derivative is ln of 10 times a to the x, and that’s it. That the derivative. Let’s take a look at a slightly harder example.

Now this example gives me two things I want to talk about. First of all, 5 times 2 to the x is not the same as 10 to the x. You have to be careful about that, remember your properties of exponents. You can’t really combine these together. It is true that 5 to the x times 10 to the x is 2 to the x. But that’s not what we have here. So just one thing I wanted to mention.

The second thing is here’s an opportunity to use the constant multiple rule. So when I take the derivative, I’m differentiating 5 times 2 to the x. And 2 to the x is an exponential function, but 5 is just a constant. By the constant multiple rule, I can pull that out. And then I’m left with this derivative of a simple exponential function. And that’s going to be 5 times ln of 2, ln of the base, times 2 to the x. And that’s it.

And then h(x), x³ plus 3 to the x. So here I want to talk about two things as well. Here I’m going to use the sum rule for derivatives, but I also want to point out that these two functions are not the same. This one is exponential. This is a power function. Now they use the same letters, they are just interchanged.

A power function has the variable in its base. An exponential function has the variable in its exponent. Make sure you use the appropriate rule for differentiation for the appropriate function. Each prime of x is the derivative with respect to x of x³ plus 3 to the x.

So I’m going to use the sum rule; the derivative of x³ plus the derivative of 3 to the x. For the power function, you use the rule that says you pull the power out in front and you replace it with one less. So this is 3x squared. But for exponential functions its ln of 3, times 3 to the x. Remember try not to confuse power functions with exponential functions when you are differentiating.

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