A dark spacetime grid stretches silently across a cosmic void. As a glowing sphere glides through it, the grid bends and ripples around its path, visualizing the curvature of spacetime from Einstein’s General Relativity. Subtle 3D rotation and faint starlight enhance the scene’s depth, revealing gravity as the graceful warping of the universe’s fabric.
Derivative of \(\mathsf { x^{n} }\) using the First Principle
Let y = \(\mathsf {x^{n} }\) ∴ y + δy = \(\mathsf { {(x + δx)^{n}} }\) ∴ δy = y + δy – y = \(\mathsf { (x + δx)^{n} }\) – \(\mathsf { x^{n} }\) or δy = \(\mathsf { [\text{ }^{n}C_0 x^{n}{(δx)}^{0} }\) + \(\mathsf {\text{ }^{n}C_1 x^{n-1}{(δx)}^{1}}\) +…
Derivative of \({e}^x\) using First Principle
Derivative of \({e}^x\) using the First Principle Let \(y\) = \({e}^x\)∴ \(y + δy\) = \({e}^{x + δx}\)∴ \(δy\) = \({e}^{x + δx}\) – \({e}^x\)or \(δy\) = \({e}^{x}\) . \( [ {e}^{δx} – 1 ]\)Dividing each side by δx </h3>or \(\dfrac {δy}{δx}\) = \(…
Derivative of sinθ using the First Principle
Derivative of \( sinθ \) using the First Principle Let \(y\) = \( sinθ \) ∴ \(y + δy\) = \( sin(θ + δθ) \) ∴ \(δy\) = \( sin(θ + δθ) \) – \( sinθ \)From Trigonometry , we have \( sin(A-B) \) = 2.\( sin \dfrac {(A-B)}{2} \).\( cos \dfrac {(A+B)}{2} \)Using the…
Derivative of cosθ using the First Principle
Derivative of \( cosθ \) using the First Principle Let \(y\) = \( cosθ \) ∴ \(y + δy\) = \( cos(θ + δθ) \) ∴ \(δy\) = \( cos(θ + δθ) \) – \( cosθ \)From Trigonometry , we have \( cos(A-B) \) = -2.\( sin \dfrac {(A+B)}{2} \).\( sin \dfrac {(A-B)}{2} \)Using the…
De Morgan’s laws
De Morgan’s First Law The complement of the union of two sets is equal to the intersection of their complements i.e. (A ∪ B )’ = A’ ∩ B’ De Morgan’s Second Law The complement of the intersection of two sets is equal to the union of their complements…
Cartesian Product of Sets
A cartesian product between two sets is defined as the set consisting of all possible ordered pairs that can be formed by taking one element from each of the sets at a given time. If A and B are two sets such that a ∈ A and b ∈ B, then the cartesian product between A…
Tuples
In mathematics, a tuple or a sequence is a list of objects arranged in an order. Such a list may have repeated objects but the order is more important. Such sequences or tuples are denoted as ( t1, t2, t3, …., tn ) where tn is the nth element of the list. Below are a…
Ordered Pairs
An ordered pair is a 2-tuple formed by taking two elements (generally numbers but can be alphabets, characters, words, or symbols). The general form of representation is (a, b) where a and b represent two distinct objects. The important thing with ordered pairs is…
Cartesian Product
The cartesian product of two sets A and B is defined as a set formed by all the possible ordered pairs of elements from A and B, such that the first element comes from set A and the second element comes from set B. The cartesian product is denoted as A × B….
Intersection operation on two sets
The intersection of two sets A & B is defined as a set that contains only those members which are common to both A and B. The intersection operation is denoted by the symbol ∩. Remember, for two disjoint sets (sets having no common elements), the…