Expert answers to "Check whether the given function is satisfying the..."

ka1leE

Answered question

2021-09-08

Check whether the given function is satisfying the Fubini’s theorem about ‎mixed partial derivative ‎
$f (x, y) = 3 x^{2} \sin (4 x y)$

Answer & Explanation

Alara Mccarthy

Skilled2021-09-09Added 85 answers

Possible derivation:
$\frac{d}{d x} (3 x^{2} \sin (4 x y))$
Factor out constants:
$= 3 (\frac{d}{d x} (x^{2} \sin (4 x y)))$
Use the product rule, $\frac{d}{d x} (u v) = v \frac{d u}{d x} + u \frac{d v}{d x}$ , where $u = x^{2} and v = \sin (4 x y) :$
$= 3 x^{2} (\frac{d}{d x} (\sin (4 x y))) + (\frac{d}{d x} (x^{2})) \sin (4 x y)$
Using the chain rule, $\frac{d}{d x} (\sin (4 x y)) = \frac{d \sin (u)}{d u} \frac{d u}{d x}$ , where $u = 4 x y$ and $\frac{d}{d u} (\sin (u)) = \cos (u) :$
$= 3 ((\frac{d}{d x} (x^{2})) \sin (4 x y) + \cos (4 x y) (\frac{d}{d x} (4 x y)) x^{2})$
Factor out constants:
$= 3 ((\frac{d}{d x} (x^{2})) \sin (4 x y) + 4 y (\frac{d}{d x} (x)) x^{2} \cos (4 x y))$
The derivative of x is 1:
$= 3 ((\frac{d}{d x} (x^{2})) \sin (4 x y) + 14 x^{2} y \cos (4 x y))$
Use the power rule, $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , where n = 2.
$\frac{d}{d x} (x^{2}) = 2 x :$
Answer:

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