Differentiation for class 11th || Calculus part 01 || Mathematical Tool

Differentiation for class 11th || Calculus part 01 || Mathematical Tool ‪@Integralganit‬
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👉 *Limits*: The concept of limits allows us to study the behavior of functions as the input values approach a specific point.
👉*Derivatives*: Derivatives measure the rate of change of a function with respect to a variable. In kinematics, derivatives are used to find the velocity and acceleration of an object.
👉*Differentiation*: Differentiation is the process of finding the derivative of a function. It is used to find the rate of change of displacement, velocity, and acceleration.
👉*Derivative Rules*: Various rules like the power rule, product rule, and quotient rule are used to differentiate functions.
👉*Geometric Interpretation*: The derivative of a function can be interpreted geometrically as the slope of the tangent to the curve at a point.
👉*Physical Interpretation*: In kinematics, the derivative of displacement gives velocity, and the derivative of velocity gives acceleration.
💡These mathematical tools are essential for understanding the concepts of kinematics, such as motion, velocity, and acceleration, and are crucial for solving problems in physics and other fields
Infinite Series Differentiation:
Term-by-Term Differentiation of Infinite Series
Differentiation of Power Series
Infinite Series Differentiation Formula
Calculus:
Differentiating Infinite Series
Term-wise Differentiation of Infinite Series
Logarithmic Series Differentiation:
Differentiation of Logarithmic Functions
Logarithmic Differentiation Formula
Calculus:
Differentiating Logarithmic Series
Chain Rule for Logarithmic Differentiation
Differentiation of Logarithmic Functions using Chain Rule
Here are some key points about logarithmic differentiation and differentiating infinite series
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate large functions
It uses logarithms and the chain rule of differentiation
It is mainly used to differentiate functions of the form f(x)g(x)
It is useful for differentiating functions that are a product of multiple sub-functions
or if one function is divided by another function
or if a function is an exponent of another function
The formula for logarithmic differentiation is: d/dx (log f(x)) = f '(x)/f(x)
Differentiating Infinite Series
Infinite series can be differentiated term by term
The theorem states that if a sequence of functions {f_n} is differentiable on [a, b] and converges uniformly on [a, b] then the derivative of the sum of the series is equal to the sum of the derivatives of the individual terms
The theorem can be used to prove that a given infinite series can be differentiated term by term
However
the converse of the theorem is not always true
and it is not possible to differentiate an infinite series term by term in all cases
Differentiating an infinite series can be used to find the derivative of a function that is defined as an infinite sum
Here are the methods to differentiate logarithmic series and infinite series
Logarithmic Series:
Take the natural logarithm of both sides of the equation.
Use the chain rule to differentiate the logarithm
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