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

\[ \boxed{\dfrac{df}{dx} = \dfrac{dv}{dt} \cdot \dfrac{dt}{dx}} \]

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.

This method is especially useful when:

• both the base and exponent contain variables;
• the function contains complicated products, quotients, or powers.

General Form:

\[ \boxed{\dfrac{dy}{dx} = y\left[\dfrac{v(x)}{u(x)}u'(x) + v'(x)\log[u(x)]\right]} \]

Since \[ y = [u(x)]^{v(x)}, \]

we may also write

\[ \boxed{\dfrac{dy}{dx} = [u(x)]^{v(x)}\left[\dfrac{v(x)}{u(x)}u'(x) + v'(x)\log[u(x)]\right]} \]

Implicit Function

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.

Explicit Function

If a relation between x and y can be easily solved for y and written as \[ y = f(x), \] then y is given as an explicit function of x.

When x and y are expressed separately as functions of the same third variable t, i.e. \[ x = f(t), \qquad y = g(t), \] the equations are called parametric equations, and t is called the parameter.

The parameter may also be denoted by \[\theta, u,\] etc.

Important Condition

The formula \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \] is directly applicable when

\[ \boxed{\frac{dx}{dt} \neq 0.} \]

If \[ \frac{dx}{dt} = 0, \] the point must be examined separately. It may correspond to a point where the tangent is vertical.

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.

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

• 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