Introduction
Differential calculus is a theory which has its origin in the solution of two old problems- one of drawing a tangent line to a curve and the other of calculating the velocity of non-uniform motion of a particle. In both the problems, the curves involved are continuous curves and the process used is the limiting process.
So the objects of study in the differential calculus are continuous functions. These problems were solved in a certain sense by Isaac Newton (English, 1642-1727) and Gottfried Willhelm Leibnitz (German, 1646-1716) and in the process differential calculus is discovered.
Increment
A small change in the value of a variable is known as the increment. A change in the value of the variable may increase or decrease. But in either case, we use the term increment to denote the change. As for example, if the value of the variable changes from 2.0 to 2.0004 or from 2.0004 to 2.0 the change known as the increment is 0.0004.
Let y = f(x) be the function of x. If the value of x changes, the value of y will also change accordingly. Generally we denote by Δx (read as ‘delta x’) to change in x and Δy the corresponding change in y.
If x changes to x + Δx, y will change to y + Δy and f(x) changes to f(x + Δx) that is, f(x) = 2x + 3 then f(x + Δx) = 2(x + Δx) + 3
Rules Of Differentiation
Here in this chapter, we’ll learn about different rules of differentiation. We’ll need these rules while solving the problems given in the exercises. So let’s have a look at the different rules of differentiation.
- The sum rule
- The product rule
- The power rule
- The quotient rule
- The chain rule
Exercise – 16.1
Introduction
Differential calculus is a theory which has its origin in the solution of two old problems- one of drawing a tangent line to a curve and the other of calculating the velocity of non-uniform motion of a particle. In both the problems, the curves involved are continuous curves and the process used is the limiting process.
So the objects of study in the differential calculus are continuous functions. These problems were solved in a certain sense by Isaac Newton (English, 1642-1727) and Gottfried Willhelm Leibnitz (German, 1646-1716) and in the process differential calculus is discovered.
Increment
A small change in the value of a variable is known as the increment. A change in the value of the variable may increase or decrease. But in either case, we use the term increment to denote the change. As for example, if the value of the variable changes from 2.0 to 2.0004 or from 2.0004 to 2.0 the change known as the increment is 0.0004.
Let y = f(x) be the function of x. If the value of x changes, the value of y will also change accordingly. Generally we denote by Δx (read as ‘delta x’) to change in x and Δy the corresponding change in y.
If x changes to x + Δx, y will change to y + Δy and f(x) changes to f(x + Δx) that is, f(x) = 2x + 3 then f(x + Δx) = 2(x + Δx) + 3
Derivatives of the Trigonometric Functions
This specific exercise is all about derivatives of trigonometric functions. In the previous exercise we study about the ways of finding derivative of a function using product rule, sum rule, power rule, quotient rule and the chain rule. But in this exercise, we won’t talk about those things. However we may need to use some of the formula of previous exercise while solving problems of this exercise too.
Some of the formula for the derivatives of trigonometric functions used in this exercise are given below. Try to remember them so that it will be easy while solving the problems given in the exercise book.
Exercise – 16.2
Introduction
Differential calculus is a theory which has its origin in the solution of two old problems- one of drawing a tangent line to a curve and the other of calculating the velocity of non-uniform motion of a particle. In both the problems, the curves involved are continuous curves and the process used is the limiting process.
So the objects of study in the differential calculus are continuous functions. These problems were solved in a certain sense by Isaac Newton (English, 1642-1727) and Gottfried Willhelm Leibnitz (German, 1646-1716) and in the process differential calculus is discovered.
Increment
A small change in the value of a variable is known as the increment. A change in the value of the variable may increase or decrease. But in either case, we use the term increment to denote the change. As for example, if the value of the variable changes from 2.0 to 2.0004 or from 2.0004 to 2.0 the change known as the increment is 0.0004.
Let y = f(x) be the function of x. If the value of x changes, the value of y will also change accordingly. Generally we denote by Δx (read as ‘delta x’) to change in x and Δy the corresponding change in y.
If x changes to x + Δx, y will change to y + Δy and f(x) changes to f(x + Δx) that is, f(x) = 2x + 3 then f(x + Δx) = 2(x + Δx) + 3
Derivatives of Exponential and Logarithmic Functions
This specific exercise is all about derivatives of exponential and logarithmic functions. In the previous exercise we study about the ways of finding derivative of a function using product rule, sum rule, power rule, quotient rule and the chain rule. We also learn about the methods of finding derivative of trigonometric functions. But in this exercise, we won’t talk about those things. However we may need to use some of the formula of previous exercise while solving problems of this exercise too.
Some of the formula for the derivatives of exponential and logarithmic functions used in this exercise are given below. Try to remember them so that it will be easy while solving the problems given in the exercise book.
