Paper 4: Fundamentals of Business Mathematics & Statistic

3.30 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Theorem. If y = f(z) and z = φ (x) then dy dy dzdx dz dx=. (proof is not shown at present)

Corr. If u = f(v), v = φ (w), w = ψ (x) then du du dv dwdx dv dw dx=..

Example 62 : To find dydx for y = 2z^2 + 1, z = 4x – 2

Solution:

Now ddzy=4z and ddxz=^4

dy dy dz. 4z. 4 16z 16(4x 2) 64x 32.

dx dz dx= = = = − = −

Rule 1.If y = ax + b, to find dydx. Let y = z, and z = ax + b.

So y = f (z) and z = f (x)

∴ ddx dz dxy dy dz=. 1=. (a. 1 + 0) = a

Rule. 2. If y = (ax + b)n, to find dydx Let y = zn and z = ax + b

Now dydz=n.zn 1− and dzdx=a

∴ dy dy dzdx dz dx=. nz= n 1− a = na (ax + b)n – 1

Example 63 : If {y = (2x + 5)^4 ] Let y = z^4 , where z = 2x + 5.

Now dzdy=4z ,^3 dxdz=^2

Now dx dz dxdy dy dz=. 4z. 2 4.2 2x 5 8 2x 5=^3 = ( + )^3 = ( + )^3

Rule 3.If y = log u (u is a function of x), then to find dydx; dy dy dudx du dx=.