How do you differentiate f(x)=x / cot (x) + 3f(x)=xcot(x)+3 using the quotient rule?

1 Answer
May 28, 2016

\frac{x+\sin ^2t(x)\cot (x)}{\sin ^2\(x)\cot ^2\(x)}x+sin2t(x)cot(x)sin2(x)cot2(x)

Explanation:

\frac{d}{dx}(\frac{x}{\cot (x)}+3\)ddx(xcot(x)+3)

Applying sum/difference rule,\(f\pm g\)^'=f^'\pm g^'

=\frac{d}{dx}\(\frac{x}{\cot t(x)})+\frac{d}{dx}(3)

Now,
=\frac{d}{dx}\(\frac{x}{\cot t(x)})
Applying quotient rule,(\frac{f}{g})^'=\frac{f^'\cdot g-g^'\cdot f}{g^2}

=\frac{\frac{d}{dx}(x)\cot (x)-\frac{d}{dx}(\cot (x))x}{\cot ^2(x)}

And,we know,\frac{d}{dx}(x)=1 and, \frac{d}{dx}(\cot (x))=-\frac{1}{\sin ^2(x)}

Also,\frac{d}{dx}(3)=0

Finally,
=\frac{\sin ^2(x)\cot(x)+x}{\sin ^2\(x)\cot ^2(x)}+0

Simplifying it,
\frac{x+\sin ^2t(x)\cot (x)}{\sin ^2\(x)\cot ^2\(x)}