Find the derivative of the function?

Question

y= 2xsinx-sinx and 
f(t) = t²/³ - t¹/³ + 4

Answer 1

The function y = 2x sinx - sinx

Apply derivative on each side with respect to x.

Apply chain rule in derivatives.

d/dx(y) = d/dx(2x sinx - sinx)

Apply product rule derivatives d/dx(uv) = uv' + vu'

u = x, v = sinx

u' = 1, v' = cosx

y' = 2(x cosx + sinx ) - cosx

y' = 2x cosx + 2 sinx - cosx

y' = (2x - 1 )cosx + 2 sinx.

Answer 2

The function f(t) = t^2/3 - t^1/3 + 4

Apply derivative on each side with respect to t.

Recall the formulae : d/dx(xⁿ) = nx^n-1

d/dx (constant) = 0

f'(t) = (2/3)t^{(2/3) - 1} - (1/3)t^(1/3)-1

f'(t) = (2/3)t^(-1/3) - (1/3)t^(-2/3)