In this chapter, we begin our study of differential calculus.

1. In this chapter, we begin our study of differential calculus. • This is concerned with how one quantity changes in relation to the changes in another quantity. DERIVATIVE is a measure of how fast does a function change in response to changes in independent variable;

2. The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool. The derivative measures the instantaneous rate of change of the function, as distinct from its average rate of change.

3. Definition of INFINITESIMAL 1:taking on values arbitrarily close to but greater than zero 2:immeasurably or incalculably small <an infinitesimal difference> Now let us observe:

4. Gradient/slope of secant and tangent line A secant is a line that intersect a curve. A tangent to a curve at a specific point is a straight line that touches the curve at that point. = Slope of the secant line joining and T f (a+Δx) - f(a) P Let us consider Δ x intervals that are getting smaller and smaller. As point T approaches point P, secant line becomes tangent line.

5. = Let =1. As end point (2,1) approaches point (1,-2) secant line approaches tangent line at (1,-2) Slopes of the secant lines connecting point (1,-2) and other points on the graph are approaching the slope of the line tangent at (1,-2). In that process both, Δf andΔxare becoming infinitesimally small approaching zero, but their ratio is approaching definite value = slope of tangent line at (1,-2) = 2

6. f (x) x Df = f (a+Δx) - f(a) POINTS: Dx S Q T R f (a+Δx) - f(a) P D x Slopes of the secant lines connecting point P and other points on the graph are approaching the slope of the line tangent at P as T approaches P. In that process both, Δf andΔxare becoming infinitesimally small approaching zero, but their ratio is approaching definite value = slope of tangent line at P

8. Definition of Derivative Very often we write instead of and (a) Derivative of the function at a fixed point : Let’s assume that can take any value of on an open interval The derivative of is the function whose value at any is provided that this limit exists. If this limit exists for each x in an open interval , then we say that fis differentiable on(has a derivative everywhere in its domain).

9. Graphycally: If function has derivative at every point in the domain it is differentiable on the domain.

10. example: Equation of the tangent line to a function at the point

11. Equivalence of Leibnitz’s and Newton’s definition of derivative Physical: rate of change Geometrical: Slope of the Tangent to a Curve The derivative of a function with respect to is defined as The slope of the tangent to a curve with respect to is defined as The derivative measures the instantaneous rate of change of the function, , as distinct from its average rate of change, . P P f (x) f (x) x x In differentiation Leibniz used the symbols and to represent "infinitely small" (or infinitesimal) increments of and , just as and represent finite increments of and .[ The ratio of two finite increments becomes the ratio of two infinitesimal increments in the process of finding the limit, yet still that ratio is a finite value. And it is equal to the slope of tangent line.

12. There are many ways to write and read the derivative of Notation: When derivatives are taken with respect to time, they are often denoted using Newton's dot notation

13. Alternate limit form of derivative

14. Differentiability A function is differentiable onan open interval (has a derivative) if the limit exists for • Functions on closed intervals must have one-sided derivatives defined at the end points. To be differentiable, a function must be continuous and smooth. • Derivatives will fail to exist at: A function has derivative at a point, if the left derivative is equal to the right derivative at that point. The left slope must be equal to the right slope. cusp corner vertical tangent discontinuity

15. Differentiability Implies Continuity If fis differentiable at x = c, then f is continuous at x= c. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous. The converse: "If a function is continuous at c, then it is differentiable at c," - is not true. This happens in cases where the function "curves sharply." Differentiability implies continuity, continuity doesn’t imply differentiability.

16. The derivative is defined at the end points of a function on a closed interval. example: The derivative is the slope of the original function.

17. Rate of change The derivative is the instantaneous rate of change of a function with respect its variable. This is equivalent to finding the slope of the tangent line to the function at a point. This is where Newton’s and Leibnitz approach meet. • Rate of change is large when the derivative is large (and therefore the curve is steep, as at the point P in the figure), the y-values change rapidly. • Rate of change is small when derivative is small, the curve is relatively flat and the y-values change slowly.

18. Although you are going to use certain simple rules to compute derivatives that will greatly simplify the task of differentiation, sometimes there will be a question to find derivative directly from the definition (very good to remind you of the true meaning of derivative). Before we introduce these simple rule without which calculus BC would be even worse nightmare, there would be no Ipad and the life as we know it ……. let us use definition of derivative, and find some simple rules.

19. Equation of the tangent line to a function at the point Equation of the normal line to a function at the point

20. Find an equation of the tangent line to the hyperbola at the point (3, 1). • The slope of the tangent at (3, 1) is: Eq. of the tangent at the point (3, 1) is The hyperbola and its tangent are shown in the figure How do you find normal at (3,1) ?

21. We can estimate the value of the derivative at any value of x by drawing the tangent at the point and estimating its slope. For instance, for x = 5, we draw the tangent at P in the figure and estimate its slope to be about 3/2, so . This allows us to plot the point on the graph of f’ directly beneath P. Repeating this procedure at several points, we get the graph shown in this figure. Tangents at x = A, B, and C are horizontal. • So, the derivative is 0 there and the graph • of f’ crosses the x-axis at those points. Between A and B, the tangents have positive slope. So, f’(x) is positive there. Between B and C, and the tangents have negative slope. So, f’(x) is negative there.

22. Derivative of a function at a point gives • The slope of the tangent line at that point • The instantaneous rate of change at that point Application of the Derivative to Motion Let be a position function as a function of time and be a velocity function as a function of time. Then: The average rate of change of position on a time interval from to is called average velocity. The instantaneous rate of change of position at time is called instantaneous velocity.