DIFFERENTIATION & INTEGRATION CHAPTER 4

 Differentiation is the process of finding the derivative of a function.  Derivative of INTRODUCTION TO DIFFERENTIATION

 Derivative of a Power Function Theorem 1  Derivative of a Constant Times a Function TECHNIQUES OF DIFFERENTIATION

 Derivative of Sum and Difference Rules

 Example :

 Derivative of a Product

 Example : Differentiate

 Derivative of Quotient

 Example : Differentiate

EXERCISE 1

 Derivative of Trigonometric Functions

 Example : Differentiate :

 Derivative of Exponential and Logarithmic Functions

 Example :

 The Chain Rule  Example :

EXERCISE 2

 Implicit Differentiation Implicit differentiation is the process of taking the derivative when y is defined implicitly or in y is a function of x.  Example :

EXERCISE 3

 A parametric derivative is a technique for finding derivative when both x and y variables depend on an independent third variable, t (time). PARAMETRIC DIFFERENTIATION

Example : Solve for if:

 Consider y =f(x) with point for function of a graph.  Tangent line – straight line that touches y = f(x).  Normal line – line that is perpendicular to tangent line.  Slope of tangent line,  Equation of the line tangent  Equation of normal line TANGENT AND NORMAL LINE

 Example : Given a curve equation at point (2,-1). a)Implicitly differentiate the equation.

b) Find the equation of the line tangent of the curve. Step 1 : Find the slope of tangent, m. Step 2 : Find the equation using

c) Find the equation of the normal line of the curve.

1.If, find the equation of the tangent line at (-1,4). 2.If, find the slope of the tangent line to the curve where. Find the equation of the line tangent to the curve where EXERCISE 4