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.

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

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

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

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

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

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.

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)`

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 \[\frac{\mathrm{d}y}{\mathrm{d}x}\neq0\].

If differentiation of an expression is done after taking the logarithm on both sides, then it is called logarithmic differentiation. Generally, we apply this method when the given expression is in one of the following forms:

1. product of a number of functions,
2. a quotient of functions,
3. a function which is the power of another function, i.e., \[[f(x)]^{g(x)}\]

• If y = f(x), then \[\frac{dy}{dx}\] = f′(x) is called the first-order derivative.
• The derivative of the first derivative is called the second-order derivative:
  \[\frac{d^2y}{dx^2}\] = f″(x)
• Higher order derivatives are written as:
  fⁿ(x) or \[\frac{d^ny}{dx^n}\]

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