Confused on reasoning for some problems

[mathtwit]

Find dy/dx
\(\displaystyle \huge y = \cos(x) \tan(x) + \sin(x) \cot(x)\)

Instructor simply notes the solution as " = sin(x) + cos(x)" and reduces to cos(x)-sin(x)

By what rule did he get rid of the tan and cot in the problem?

Next one

Find dy/dx
\(\displaystyle \huge y=\frac{3c}{2x^5}\)

His reasoning is this:

\(\displaystyle \huge y' = \frac{0 * 2x^5 - 3c * 10x^4}{(2x^5)^2}\)

That's not how I learned the quotent rule, and where's this zero coming from?

I'm mainly interested in what steps I'm missing.

 

[galactus]

The first one is just a matter of manipulating the identity
\(\displaystyle \L\\\frac{sin(x)}{cos(x)}=tan(x)\).

\(\displaystyle \L\\\frac{sin(x)}{cos(x)}=tan(x)\), therefore, \(\displaystyle \L\\cos(x)tan(x)=sin(x)\)

\(\displaystyle \L\\sin(x)cot(x)=\frac{sin(x)}{tan(x)}=cos(x)\)

So you have \(\displaystyle \L\\sin(x)+cos(x)\). Now, differentiate.


For #2. c is a constant, since you're differentiating with respect to x.

Therefore, 3c is a constant. The derivative of a constant is 0.

\(\displaystyle \L\\\frac{2x^{5}(0)-3c(10x^{4})}{4x^{10}}=\frac{-15c}{2x^{6}}\)

 

[mathtwit]

Brilliant, thanks!