00:01
So in math applications, we often use exponential functions, trigonometric functions.
00:10
So for that reason, it's helpful for us to be able to essentially take the derivative of those.
00:17
When we're in calculus, the derivative is such a crucial part of understanding rates of change and all those things that we want to be able to use logarithms, exponents, trigonometric functions, and be able to take their.
00:30
Derivatives.
00:32
So one way in which we do this is if we have a complicated function such as this one right here, where we have an exponential function, but the variable is in both the base and the exponent.
00:51
So the only way that we solve this is taking the natural log of both sides.
00:54
When we do that, this cosine x is now a power, so it can move out in front and multiply.
01:03
Then the nableness.
01:04
Log of y now we can perform implicit differentiation so what we'll have is the d d x of the natural log of y is going to be one over y times y prime and then uh cosine of x natural times natural log of x the derivative of that will be cosine of x times one over x plus natural log of x times a negative sine x...