The first example is from: Widder, David V., "Advanced Calculus", Dover Books on Mathematics
Prerequisites: single-variable calculus, specifically u-substitution
Example 1:
Find \frac{\partial }{\partial x}\frac{sin(xy)}{cos(x+y)}
Solution:
We will need the quotient rule: \left ( \frac{f}{g} \right )' = \frac{f(x)g'(x) - f'(x)g(x)}{[g(x)]^{2}}
Setting f(x) = sin(xy)
Using the rule: \frac{\mathrm{d} }{\mathrm{d} x}sin(u(x)) = cos(u(x))\frac{\mathrm{d} u}{\mathrm{d} x}
And conversely, setting g(x) = cos(x+y)
Using the rule: \frac{\mathrm{d} }{\mathrm{d} x}cos(u(x)) = -sin(u(x))\frac{\mathrm{d} u}{\mathrm{d} x}
Putting the parts together: \frac{\partial }{\partial x}\frac{sin(xy)}{cos(x+y)} = \frac{sin(xy)(-sin(x+y))-(ycos(xy)cos(x+y))}{[cos(x+y)]^{2}}
This is usually sufficient for most professors. It is assumed that if you can get this far, you can simplify.
No comments:
Post a Comment