Revision: Differentiation Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

If \(u = g(x)\) and \(y = f(u)\), then \(y = f(g(x))\) is called a composite function. Here, \(g(x)\) is the inner function and \(f(u)\) is the outer function.

Let \[f\] be a real-valued function which is a composite of two functions \[u\] and \[v\]; i.e., \[f = v \circ u\]. Suppose \[t = u(x)\] and if both \[\frac{dt}{dx}\]and \[\frac{dv}{dt}\]exist, we have

If a function reverses the action of another function, it is called its inverse function. For example, if \[y = \sin^{-1} x\], then \[x = \sin y\], which means the inverse function converts a trigonometric value back into an angle.

If differentiation of a function is performed after taking logarithm on both sides, the process is called logarithmic differentiation.

Implicit differentiation means differentiating both sides of an equation with respect to x, while remembering that y depends on x. Therefore, whenever a term containing y is differentiated, the factor \[\frac{dy}{dx}\] appears by the chain rule.

When x = f(t) and y = g(t), the relation between x and y is said to be in parametric form.

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)`

• A composite function has one function inside another function.

• The chain rule formula is \[\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}\]

• First differentiate the outer function, then multiply by the derivative of the inner function.

• The derivative of an inverse function is usually found using implicit differentiation.

• For \[\sin^{-1} x\] and \[\cos^{-1} x\], the denominator is \[\sqrt{1 - x^2}\].

• For \[\tan^{-1} x\] and \[\cot^{-1} x\], the denominator is \[1 + x^2\].

• For \[\sec^{-1} x\] and \[\csc^{-1} x\], the denominator involves \[|x|\sqrt{x^2 - 1}\].

• Negative signs are especially important in \[\cos^{-1} x\], \[\cot^{-1} x\], and \[\csc^{-1} x\].

• Domain restrictions must be checked before applying formulas.

• Use logarithmic differentiation when the function is a complex product, quotient, or variable exponent form.

• Write y = function first, then take \[\ln\] on both sides.

• Apply logarithmic rules before differentiating.

• Differentiate \[\ln y\] carefully: \[\frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx}\].

• Substitute the original value of y at the end.

• Ensure the expression inside logarithm remains positive.

• If an equation contains both x and y and cannot be solved directly for y, it is called an implicit function.
• Implicit functions are generally written in the form:
  f(x, y) = 0
• To differentiate an implicit function, differentiate both sides with respect to x, treating y as a function of x.

• Parametric form means both x and y are written in terms of a third variable.

• The third variable is called the parameter.

• The main formula is:

  \[\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\]

• This formula is based on the chain rule.

• Always check that \[\frac{dx}{dt} \neq 0\].

• The final answer may remain in terms of the parameter unless the question asks for conversion.

• 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