Revision: Differentiation Maths HSC Commerce (English Medium) 12th Standard Board Exam Maharashtra State Board

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then

\[\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}\]

If y = f(x) is a differentiable function of  x such that the inverse function x = f ^{− 1}(y) exists, then x is a differentiable function of 
y and

\[\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}},\frac{dy}{dx}\neq0\]

If x = f(t) and y = g(t) are differential functions of parameter ‘t’, then y is a differential function of x and 

\[\begin{aligned}
\frac{dy}{dx} & =\frac{\frac{dy}{dt}}{\frac{dx}{dt}}, \\
 \\
\frac{dx}{dt} & \neq0
\end{aligned}\]

If y = f(x) is a differentiable function of x, then its derivative f′(x) is also a function of x.

If this derivative f′(x) is again differentiable, its derivative is called the second derivative of f(x).

\[f^{\prime\prime}(x)\quad\mathrm{or}\quad\frac{d^2y}{dx^2}\]

If the second derivative is differentiable, its derivative is called the third derivative, denoted by:

\[f^{\prime\prime\prime}(x)\quad\mathrm{or}\quad\frac{d^3y}{dx^3}\]

Continuing this process, the derivative obtained after differentiating f(x) n times is called the nth derivative of f(x), and is denoted by:

\[f^{(n)}(x)\quad\mathrm{or}\quad\frac{d^ny}{dx^n}\]

These derivatives beyond the first derivative are called higher-order derivatives.

1. Sum Rule:

\[y=u\pm v\] then \[\frac{dy}{dx}=\frac{du}{dx}\pm\frac{dv}{dx}\]

2. Product Rule:

\[y=uv\] then \[\frac{dy}{dx}=u\frac{d\nu}{dx}+\nu\frac{du}{dx}\]

3. Quotient Rule:

\[y=\frac{u}{v}\] where v ≠ 0 then \[\frac{dy}{dx}=\frac{\nu\frac{du}{dx}-u\frac{d\nu}{dx}}{\nu^{2}}\]

4. Difference Rule:

y = u − v then \[\frac{dy}{dx}=\frac{du}{dx}-\frac{dv}{dx}\]

5. Constant Multiple:

y = k. u then \[\frac{dy}{dx}=k.\frac{du}{dx}\], k constant.

If: u = f(x),v = g(x)

Then: \[\frac{du}{dv}=\frac{du/dx}{dv/dx}\]

General implicit form: F(x,y) = 0

\[x^my^n=(x+y)^{m+n}\]

\[\frac{dy}{dx}=\frac{y}{x}\]

Given, y = 5 cos x – 3 sin x

Differentiating both sides with respect to x,

`dy/dx = 5 d/dx cos x - 3 d/dx sin x`

= 5 (−sin x) − 3 cos x

= −5 sin x − 3 cos x

Differentiating both sides again with respect to x,

`(d^2 y)/dx = - 5 d/dx sin x - 3 d/dx cos x`

= −5 cos x − 3 (−sin x)

= 3 sin x − 5 cos x

Hence, `(d^2 y)/dx^2 + y` = 0

(3 sin x − 5 cos x) + (5 cos x − 3 sin x) = 0 ...(On substituting the value of y)

If y is a differentiable function of u and u is a differentiable function of x, then

\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\cdot\frac{\mathrm{d}u}{\mathrm{d}x}\]

If y = f(x) is a differentiable function of x such that the inverse function x = f⁻¹(y) exists, then x is a differentiable function of y and

\[\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{1}{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}\], where \[\frac{\mathrm{d}y}{\mathrm{d}x}\neq0\].

1. Elasticity of Demand

\[\eta=-\frac{P}{D}\cdot\frac{dD}{dP}\]

2. Marginal Revenue & Elasticity Relation

\[R_m=R_A\left(1-\frac{1}{\eta}\right)\]

3. Propensity to Consume & Save

MPC + MPS = 1

APC + APS = 1