derivatives?

Question

if anyone can help/break this down for me....THANKYOU
Find the first derivative:

1) (x^2)/tanx
2) cosx + (x^2)
3) 3cscx
4) 2cos(x^2)/(x^3)
5) x^2 + (sin4x/2)

Answer 1

For any functions f and g,

(f +/ g)' = f' +/- g'
(constant * f)' = constant * f'.

Those are trivial.

Beyond that, you should know the more general

Product Rule: (fg)' = f'g + g'f

How to handle exponents: (x^n)' = nx^(n-1)

Chain Rule: (f(g))' = f'(g) * g'

Quotient Rule: (f/g)' = (f'g - g'f)/g^2

If you can derive the Quotient Rule from the others, you're probably in pretty good shape overall.

These rules, and the knowledge that sin' = cos and cos' = -sin, is all you need to solve the problems above. In case of doubt, break things into pieces. For example, for #5 differentiate x^2 and sin4x/2 separately. As for the derivative of sin 4x/2, well that's just 1/2 the derivative of sin4x. And the Chain Rule tells us that derivative is 4cos4x, because (4x)' = 4.