This week in AP Calc we talked about finding the derivatives of exponential and logarithmic functions in preparation for our chapter test on Monday. I found finding the derivatives of exponential and logarithmic functions to actually be kind of easy, I just got a little confused sometimes when it came to simplifying. Other than that I thought they were a little bit easier to find than the derivatives of other functions.
Basically, you still follow the same chain rule method just in a slightly different way. The first rule: dy/dx e^x = e^x
is really easy to understand. If you are given a function like: f(x) = e^(x^2 - 4), you simply use the chain rule as well as the rule for e^x.
Since the derivative of e^x is equal to itself, the outside functions derivative is the same as its original functions. So, you would start with: f'(x) = e^(x^2 - 4), now you need to multiply by the derivative of the inside function. Now, you would have f'(x) = e^(x^2 - 4)[2x]
f'(x) = 2x[e^(x^2 - 4)]
The second rule was: dy/dx a^x = a^x(ln(a))
So, if you have: f(x) = 2^(x^2 + 3x), you would use this rule to find the derivative of the outside function first. Then, you would multiply by the derivative of the inside function using the chain rule.
f'(x) = 2^(x^2 + 3x)[ln(2)][2x + 3]
The third rule we learned was: dy/dx ln(x) = 1/x
If you have: f(x) = ln(x^3 + 4x), you would use this rule to find the derivative of the outside function first. Then, you would multiply by the derivative of the inside function using the chain rule.
f'(x) = 1/(x^3 + 4x) [3x^2 + 4]
which then becomes: f'(x) = 3x^2 + 4
x^3 + 4x
We also talked about finding the derivatives of inverse trig. functions which was pretty simple to understand as well. Like I said before the only thing with these that I really struggle with is understanding how to simplify accurately!
Basically, you still follow the same chain rule method just in a slightly different way. The first rule: dy/dx e^x = e^x
is really easy to understand. If you are given a function like: f(x) = e^(x^2 - 4), you simply use the chain rule as well as the rule for e^x.
Since the derivative of e^x is equal to itself, the outside functions derivative is the same as its original functions. So, you would start with: f'(x) = e^(x^2 - 4), now you need to multiply by the derivative of the inside function. Now, you would have f'(x) = e^(x^2 - 4)[2x]
f'(x) = 2x[e^(x^2 - 4)]
The second rule was: dy/dx a^x = a^x(ln(a))
So, if you have: f(x) = 2^(x^2 + 3x), you would use this rule to find the derivative of the outside function first. Then, you would multiply by the derivative of the inside function using the chain rule.
f'(x) = 2^(x^2 + 3x)[ln(2)][2x + 3]
The third rule we learned was: dy/dx ln(x) = 1/x
If you have: f(x) = ln(x^3 + 4x), you would use this rule to find the derivative of the outside function first. Then, you would multiply by the derivative of the inside function using the chain rule.
f'(x) = 1/(x^3 + 4x) [3x^2 + 4]
which then becomes: f'(x) = 3x^2 + 4
x^3 + 4x
We also talked about finding the derivatives of inverse trig. functions which was pretty simple to understand as well. Like I said before the only thing with these that I really struggle with is understanding how to simplify accurately!