Revision: Differentiation Maths and Stats HSC Science (General) 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

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

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

cos y = x cos (a + y)

∴ x = `(cos y)/(cos (a + y))`

On differentiating with respect to y,

`cos (a + y) d/dy cos y - cos y d/dy`

`therefore dx/dy = (cos (a + y))/(cos^2 (a + y))`

`= (- sin y cos (a + y) + cos y sin (a + y))/(cos^2 (a + y))`

`= (sin (a + y) cos y - cos (a + y) sin y)/(cos^2 (a + y))`

`= (sin (a + y - y))/(cos^2 (a + y))`   ...[∵ sin (A − B) = sin A cos B − cos A sin B]

`= (sin a)/(cos^2  (a + y))`

`therefore dy/dx = (cos^2 (a + y))/(sin a)`

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