Suppose you know the area of a circle and the measure of the central angle of a sector. Describe the process of finding the area
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(i) I would first suggest writing this function as a product of the functions,
then apply the product rule. Hopefully it's clear which function each of f, g, and h refer to.
We then have, using the power and chain rules,
For the third function, we first rewrite in terms of the logarithmic and the exponential functions,
Then by the chain rule,
By the product rule, we have
You could simplify this further if you like.
In Mathematica, you can confirm this by running
D[(4+3x^2)^(1/2) (x^2+1)^(-1/3) Pi^x, x]
The immediate result likely won't match up with what we found earlier, so you could try getting a result that more closely resembles it by following up with Simplify or FullSimplify, as in
FullSimplify[%]
(% refers to the last output)
If it still doesn't match, you can try running
Reduce[<our result> == %, {}]
and if our answer is indeed correct, this will return True. (I don't have access to M at the moment, so I can't check for myself.)
(ii) Given
differentiating both sides with respect to x by the quotient and chain rules, taking y = y(x), gives
which could be simplified further if you wish.
In M, off the top of my head I would suggest verifying this solution by
Solve[D[x*y[x]^3/(1 + Sec[y[x]]) == E^(x*y[x]), x], y'[x]]
but I'm not entirely sure that will work. If you're using version 12 or older (you can check by running $Version), you can use a ResourceFunction,
ResourceFunction["ImplicitD"][<our equation>, x]
but I'm not completely confident that I have the right syntax, so you might want to consult the documentation.