The Differentiation

DIFFERENTIATION : 

DIFFERENTIATION By: Manpreet Oberoi

What is Differentiation? : 

What is Differentiation? Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

Slide 3: 

The derivative is the instantaneous rate of change of a function with respect to one of its variables The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change"). This concept has many applications in electricity, dynamics, economics, fluid flow, population modelling and so on.

Why Study Differentiation? : 

Why Study Differentiation? There are many applications of differentiation in science and engineering. Differentiation is also used in analysis of finance and economics. One important application of differentiation is in the area of optimisation, which means finding the condition for a maximum (or minimum) to occur. This is important in business (cost reduction, profit increase) and engineering (maximum strength, minimum cost.)

This is equivalent to finding the slope of the tangent line to the function at a point. : 

This is equivalent to finding the slope of the tangent line to the function at a point.

Let P = (x,y) and Q := (a,b). Let : 

Let P = (x,y) and Q := (a,b). Let

Then the slope of the line : 

Then the slope of the line

Now, we chose an arbitrary interval to be Delta-x. The smaller Delta-x is, the more accurate this approximation is. : 

Now, we chose an arbitrary interval to be Delta-x. The smaller Delta-x is, the more accurate this approximation is.

What we want to do is decrease the size of Delta-x as much as possible. We do this by taking the limit as Delta-x approaches zero. In the limit, assuming the limit exists, we will find the exact slope of the tangent line to the curve at the given point. This value is the derivative; : 

What we want to do is decrease the size of Delta-x as much as possible. We do this by taking the limit as Delta-x approaches zero. In the limit, assuming the limit exists, we will find the exact slope of the tangent line to the curve at the given point. This value is the derivative;

There are a few different, but equivalent, versions of this definition. In more general considerations, h is often used in place of Delta-x. Or Delta-y is used in place of : 

There are a few different, but equivalent, versions of this definition. In more general considerations, h is often used in place of Delta-x. Or Delta-y is used in place of

This leads to three commonly used ways of expressing the definition of the derivative: : 

This leads to three commonly used ways of expressing the definition of the derivative:

When is a function differentiable? : 

When is a function differentiable? A function is differentiable when the definition of differentiation can be applied in a meaningful manner to it.

When would this definition not apply? : 

When would this definition not apply? It would not apply when the limit does not exist. Then, we want to look at the conditions for the limits to exist.

Derivatives of Polynomials : 

Derivatives of Polynomials The good news is that we can find the derivatives of polynomial expressions without using the delta method

Isaac Newton and Gottfried Leibnitz obtained these rules in the early 18th century. They followed from the "first principles" approach to differentiating, and make life much easier for us. : 

Isaac Newton and Gottfried Leibnitz obtained these rules in the early 18th century. They followed from the "first principles" approach to differentiating, and make life much easier for us. Isaac Newton Gottfried Leibnitz

Derivative of a constant : 

Derivative of a constant This is basic. In English, it means that if a quantity has a constant value, then the rate of change is zero.

n-th power of x: : 

n-th power of x: This follows from the delta method.

Constant product: : 

Constant product: Here, y is some function of x. It means that if we are finding the derivative of a constant times that function, it is the same as finding the derivative of the function first, then multiplying by the constant.

Derivative of a sum: : 

Derivative of a sum: Here, u and v are functions of x. The derivative of the sum is equal to the derivative of the first plus derivative of the second.

Derivatives of Products and Quotients : 

Derivatives of Products and Quotients

PRODUCT RULE : 

PRODUCT RULE If u and v are two functions of x, then the derivative of the product uv is given by... In words, this can be remembered as: "The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first."

We can write the product rule in many different ways: : 

We can write the product rule in many different ways: We can write the product rule in many different ways: OR  … etc. We can write the product rule in many different ways: OR  … etc. We can write the product rule in many different ways: OR

(A quotient is just a fraction.)If u and v are two functions of x, then the derivative of the quotient u/v is given by... : 

(A quotient is just a fraction.)If u and v are two functions of x, then the derivative of the quotient u/v is given by... In words, this can be remembered as: "The derivative of a quotient equals bottom times derivative of top minus top times derivative of the bottom, divided by bottom squared."

You will also see the quotient rule written as: : 

You will also see the quotient rule written as: OR

Differentiating Powers of a Function : 

Differentiating Powers of a Function Function of a Function If y is a function of u and u is a function of x, then we say, “y is a function of the function u”

Slide 26: 

Consider the function y = (5x + 7)12. If we let u = 5x + 7 (the inner-most expression), then we could write our original function as y = u12 We have written y as a function of u, and in turn, u is a function of x. This is a vital concept in differentiation, since many of the functions we meet from now on will be functions of functions, and we need to recognise them in order to differentiate them properly.

Chain Rule : 

Chain Rule To find the derivative of a function of a function, we need to use the Chain Rule: This means we need to Recognize u (always choose the inner-most expression, usually the part inside brackets, or under the square root sign). Then we need to re-express y in terms of u. Then we differentiate y (with respect to u), then we re-express everything in terms of x. The next step is to find du/dx. Then we multiply dy/du and du/dx. EXAMPLES..

The Derivative of a Power of a Function : 

The Derivative of a Power of a Function Power Rule An extension of the chain rule is the Power Rule for differentiating. We are finding the derivative of un (a power of a function): EXAMPLES..

Differentiation of Implicit Functions : 

Differentiation of Implicit Functions We meet many equations where y is not expressed explicitly in terms of x only, such as: It is usually difficult, if not impossible, to solve for y so that we can then find . We need to be able to find derivatives of such expressions to find the rate of change of y as x changes. To do this, we need to know implicit differentiation. EXAMPLES..

Higher Derivatives : 

Higher Derivatives We can continue to find the derivatives of a derivative. We find the second derivative by taking the derivative of the first derivative, third derivative by taking the derivative of the second derivative... etc

Slide 31: 

THANK YOU