Topics
Complex Numbers Old
Expansion of Sinn θ,Cosn θ in Terms of Sines and Cosines Of Multiples Of θ And Expansion of Sinnθ, Cosnθ In Powers of Sinθ, Cosθ
Separation of Real and Imaginary Parts of All Types of Functions
Circular Functions of Complex Number and Hyperbolic Functions.Inverse Circular and Inverse Hyperbolic Functions. Logarithmic Functions.
Powers and Roots of Exponential and Trigonometric Functions
Matrices and Numerical Methods Old
Solution of System Of Linear Algebraic Equations
Types of Matrices and Rank of a Matrix
Differential Calculus Old
Euler’S Theorem on Homogeneous Functions with Two and Three Independent Variables (With Proof)
Partial Differentiation
Successive Differentiation
Application of Partial Differentiation, Expansion of Functions , Indeterminate Forms and Curve Fitting Old
Fitting of Curves by Least Square Method for Linear, Parabolic, And Exponential
Maxima and Minima of a Function of Two Independent Variables
Taylor’S Theorem and Taylor’S Series, Maclaurin’S Series
Complex Numbers
- Review of Complex Numbers‐Algebra of Complex Number
- Different Representations of a Complex Number and Other Definitions
- D’Moivre’S Theorem
- Powers and Roots of Exponential Function
- Powers and Roots of Trigonometric Functions
- Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ
- Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
- .Circular Functions of Complex Number
- Hyperbolic functions of complex number
- Inverse Circular Functions
- Inverse Hyperbolic Functions
- Separation of Real and Imaginary Parts of All Types of Functions
Logarithm of Complex Numbers , Successive Differentiation
Successive Differentiation
- Successive Differentiation
- nth Derivative of Standard Functions
- Leibnitz’S Theorem (Without Proof) and Problems
Logarithm of Complex Numbers
- Logarithmic Functions
- Separation of Real and Imaginary Parts of Logarithmic Functions
Matrices
- Rank of a Matrix Using Echelon Forms
- Reduction to Normal Form
- PAQ in normal form
- System of Homogeneous and Non – Homogeneous Equations
- consistency and solutions of homogeneous and non – homogeneous equations
- Linear Dependent and Independent Vectors
- Application of Inverse of a Matrix to Coding Theory
Partial Differentiation
- Partial Derivatives of First and Higher Order
- Total Differentials
- Derivatives of Composite Functions
- Differentiation of Implicit Functions
- Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
- Deductions from Euler’S Theorem
Applications of Partial Differentiation , Expansion of Functions
- Maxima and Minima of a Function of Two Independent Variables
- Jacobian
- Taylor’S Theorem (Statement Only)
- Taylor’S Series Method
- Maclaurin’s series (Statement only)
- Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
- Binomial Series
Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
- Indeterminate Forms
- L‐ Hospital Rule
- Problems Involving Series
- Solution of Transcendental Equations
- Solution by Newton Raphson Method
- Regula – Falsi Equation
- Solution of System of Linear Algebraic Equations by Gauss Elimination Method
- Gauss Jacobi Iteration Method
- Gauss Seidal Iteration Method
Estimated time: 4 minutes
CBSE: Class 12
Definition: 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.
CBSE: Class 12
Example 1
Find \[\frac{dy}{dx}\], if \[y + \sin y = \cos x\].
Solution: We differentiate the relationship directly with respect to \[x\], i.e.,
\[\frac{dy}{dx} + \frac{d}{dx}(\sin y) = \frac{d}{dx}(\cos x)\]
which implies using the chain rule
\[\frac{dy}{dx} + \cos y \cdot \frac{dy}{dx} = -\sin x\]
This gives \[\frac{dy}{dx} = -\frac{\sin x}{1 + \cos y}\]
where \[y \neq (2n + 1)\pi\]
CBSE: Class 12
Maharashtra State Board: Class 12
Maharashtra State Board: Class 12
Key Points: Derivative of Implicit Functions
- 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.
Video Tutorials
Shaalaa.com | Differentiation - Implicit Functions
to track your progress
