BAM Mr. Tarrou! In this calculus video we're going to be taking the derivative
of y is equal to x raised to the power of sine of X if you have not seen one of
these problems that have a variable base and a variable exponent well basically
you're gonna have to figure it out because we have a rule which is the
derivative of U raised to the N power a variable base and a fixed exponent is
equal to n times u to the N minus 1 power times u Prime and we have the
derivative of a to the U where the now base is fixed as opposed to the variable
is equal to the natural log of a times a to the U times u prime basically you
need to change this base a to base e through some kind of logarithmic work
and come out with your formula here your identity your rule for taking that
derivative however here we have a verbal base and a variable exponent and that
means that we're going to need to work through this if you want to do it as
cleanly as possible unlike the video I made years ago which still involve
logarithms we're going to take the natural log of both sides use the power
property of logarithms to expand this so we don't have a function raised by
another function take the derivative implicitly and then ultimately arrive I
have an answer for y prime or dy/dx and this is what it's going to look like
take the natural log of both sides then remember that of course we get exponents
from logarithms so we basically have an exponent raised by an exponent so we're
going to use the power property of logarithms to bring this sine of X out
as a coefficient the natural log of Y is equal to the sine of X times the natural
log of X and now see we have the multiplication or the product of two
functions and then we can go back to our using that product rule for taking
derivatives I'm going to take the derivative of both sides of this
equation implicitly we're going to take the derivative with respect to X that is
our original sort of independent variable
and over here with the natural log of line we're not just going to get one
over why because we're not taking the derivative with respect to why it's with
respect to X so we have over here on the left one over Y times dy/dx and on the
right hand side using the product rule we have the sine times the derivative of
the natural log of X which is 1 over X and then plus the second factor the
natural log of x times the derivative plus no times the derivative of sine
which is cosine but we have to add there so we have first times the derivative of
the second plus the second times the derivative of the first the derivative
of sine is cosine of X and that's it we're practically done all we have to do
now is solve for y prime by multiplying both sides of the equation by Y
but we don't really want a first derivative that you know has that
original independent variable of X and also have it be in terms of Y to be like
Q's and writing the second derivative with the first derivative notation still
in our answers so we're going to take Y out and replace it with what it's equal
to which of course is X raised to the sine of X power that is all there is to
take in the derivative of a function where both the base and the exponent our
functions are variables not just where we have a verbal base and a fixed
exponent or a fixed base and variable exponent I miss a true go to your
homework
you