derivatives and their applications
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Derivatives & their Applications
Mathematics Presentations
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What is derivative?• The derivative of a function of a real variable measures
the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable).
• For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced.
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What is differentiation?• The process of finding a derivative is
called differentiation.• The reverse process is called antidifferentiation.• The fundamental theorem of calculus states that
antidifferentiation is the same as integration.• Differentiation is also known as the process to find rate
of change• Derivative tells us slope of function at any point.
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As it is also rate of change• The rate of change of a function is expressed as a ratio
between a change in one variable relative to a corresponding change in another.
• Rate of change is also given by limit value.
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Average rate of change• The Average Rate of Change is defined as the
ratio of the difference in the function f(x) as it changes from 'a' to 'b' to the difference between 'a' and 'b'. The average rate of change is denoted as A(x).
• And is given as the formula• A(x) • The Average Rate of Change Formula calculates
the slope of a line or a curve on a given range.
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Average rate of change-Ex
Calculate the average rate of change of a function, f(x) = 3x + 5 as x changes from a to b ?Solution: Given, f(x) = 3x + 5 a = 3 , b = 6Putting the valuesf(3) = 3(3) + 5 :f(6) = 3(6) + 5 f(3)=14 : f(6)=23The average rate of change is,
A(x) =
A(x) = = A(x) = 3
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Notations for derivatives Leibniz's notation. , xLagrange's notation.
f’ f”Euler's notation.
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Derivative Notation
A Function
f(x)
y
X2 + 1
Newton Form
f ’(x)
y ’
(X2 + 1)’
Leibniz Form
(f(x))
(y)
(X2 + 1)
ddx
ddx
ddx
or
or
dfdx
dfdx
Derivative Notation
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Notation
There are lots of ways to denote the derivative of a function y = f(x).
f’(x) the derivative of f the derivative of f with y’ y prime respect to x.
the derivative of y the derivative of f at x with respect to x. dxdy
dxdf
)(xfdxd
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Rules for derivatives
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Rules for derivatives
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Rules for derivatives
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Rules for derivatives
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Rules for derivatives
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Product Rule Example
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Examples
Product rule:
• This is used when differentiating a product of two functions.
Differentiate x(x² + 1)
• let u = x and v = x² + 1
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Examples
Product rule:
= Putting the values.
=(x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .
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Quotient Rule Example
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Examples
Power Rule:
"what is the derivative of ?"• We can use the Power Rule, where n=6:
• So,• = 6 = 6
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Chain rule
A special rule, the chain rule, exists for differentiating a function of another function.
In order to differentiate a function of a function, y = f(g(x)),
That is to find , we need to do two things:
1. Substitute u = g(x). This gives us y = f(u) Next we need to use a formula that is known as the Chain Rule.
2. Chain Rule
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Chain Rule Example
Chain rule:If • Find Solution:
• Taking derivative to both side -------(i
• Taking derivative to both side
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Chain rule:• Using chain rule
• = • = 12u
=12(2x-1) =24x-12
Chain Rule Example
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Chain Rule Example
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Higher derivatives. 1st ,2nd 3rd
• Any derivative beyond the first derivative can be referred to as a higher order derivative.
• The derivative of the function f(x) may be denoted by f’(x)
• Its double (or "second") derivative is denoted by f ’’(x).
• This is read as "f double prime of x," or "The second derivative of f(x)."
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Higher derivatives. Notations:Let f(x) be a function in terms of x. The following are notations for higher order derivatives.
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2nd Derivative Ex.
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Higher Order Derivative Ex.
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Partial derivatives
Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation
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Partial derivatives:
The partial derivative of f is with respect to its variable.
Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee"
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Example
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Applications of partial derivatives:
• Derivatives are constantly used in everyday life to help measure how much something is changing. They're used by the government in population censuses, various types of sciences, and even in economics..
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Applications of partial derivatives:
• Derivatives in physics.You can use derivatives a lot in Newton law of motion where the velocity is defined as the derivative of the position over time and the acceleration, the derivative of the velocity over time.
• Derivatives in chemistry.One use of derivatives in chemistry is when you want to find the concentration of an element in a product.
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Concave Up
• The derivative of a function gives the slope• When the slope continually increases, the function is
concave upward.• Taking the second derivative actually tells us if the slope
continually increases or decreases.• When the second derivative is positive, the function is
concave upward.• f ''(x) > 0 for x > 0;
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Concave Up:
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Concave down:
• When the slope continually decreases, the function is concave downward.
• When the second derivative is negative, the function is concave downward.
• f ''(x) < 0 for x < 0.
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Concave down:
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Derivatives of polynomials
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Derivatives of polynomials
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Derivative of Trigonometric functions
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Derivative of Trigonometric functions
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Derivatives of Exponential and Logarithmic Functions
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Derivatives of Exponential and Logarithmic Functions
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