Chain rule for the derivative of a composite functiony=(sinx)sqrt x.

trapskrumcu 2022-09-26 Answered
Chain rule for the derivative of a composite function
y = ( sin x ) x .
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Answers (2)

Zariah Fletcher
Answered 2022-09-27 Author has 8 answers
Use logarithmic differentiation here:
y = ( sin ( x ) ) x ln ( y ) = x ln ( sin ( x ) ) 1 y d y d x = 1 2 x 1 / 2 ln ( sin ( x ) ) + x cos ( x ) sin ( x ) .
The problem with the straight-forward approach is that you don't have a power rule, nor do you have an exponent rule you can use. It's a bit analogous to differentiating x x .
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sombereki51
Answered 2022-09-28 Author has 3 answers
It's easier if you write
y = e x ln ( sin ( x ) )
Then
y = e x ln ( sin ( x ) ) ( x ln ( sin ( x ) ) ) = ( sin ( x ) ) x ( x ln ( sin ( x ) ) )
Now you only have to differentiate x ln ( sin ( x ) ), in which case you need the product rule and the chain rule.
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