differentiate the following equations with respect to xSolut

differentiate the following equations with respect to x:

Solution

(i) y= x.e^2x

  In general, the derivative of a product (such as u*v) with respect to x is

v*du/dx + u*dv/dx (for clarity, * signifies multiplication)

In your example, u = x and v = e^(2x).

The u and v derivatives are: du/dx = 1 and dv/dx = 2xe^(2x - 1)

So, y\' = dy/dx = e^(2x)*[1] + x*[2xe^(2x - 1)]

Simplifying, y\' = e^(2x) + 2x^(2)e^(2x - 1) = e^(2x - 1)[e + 2x^(2)] or

y\' = e^(2x)[1 + [2x^(2)/e]

(ii) y= 2x³.cosx

= apply uv formulae

= 2x³.-sinx - cosx.3x²

= -x²(2xsinx+3cosx)

(iii) x²-1/sinx

apply u/v formulae,

u.v\' - v. u\'/v²

= x²-1.cosx - sinx . 2x / sin².x

= x2cosx - cosx - 2x.sinx /sin²x

(iv) y= (2x³-5x)5

y= 10x³-25x

dy/dx= 30x²- 25