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Chapter 3 - More About Derivatives
3.1 Rules of Differentiation
(1) Power Rule If is any real number and , then. Example 3.1. Find if (i) (ii) , (iii) and (iv). Solution 3.1. (2) The derivative of a constant is zero. That is if , where c is a constant, then. (3) Derivative of a constant multiple and sums of functions Let c be a constant and also let u and v be functions of x. Then (a) (b) (c). Example 3.1. (a) Differentiate the following function with respect to x. . (b) If , find the equation of the tangent line to the graph of at the point . Solution 3.1. (a) (b) (4) Product Rule Let u and v be functions of x. Let , then.
Example 3.1. Find if (a) and (b). Solution 3.1. (5) Quotient Rule Let u and v be functions of x and let. Then. Example 3.1. Find if (a) , (b) and (c). Solution 3.1. (6) The Chain Rule Suppose you were asked to differentiate the function. The rules learnt earlier would not enable you to calculate or. Observe that is a composite function. Let , then. Regard as the rate of change of w.r.t. , as the rate of change of w.r.t. and as the rate of change of w.r.t.. We get. The Chain Rule states that if and are both differentiable functions, then . Example 3.1.
Theorem 3.3.1: If , the function is a one-to-one, continuous, increasing function with domain and range R. 3.3.1 Natural Logarithms , , ,. 3.3.2 Derivatives . In general,. , and ( constants). and. Example 3.3.
. Find. Solution 3.3. Example 3.3. Differentiate the following functions: (a) , (b) , (c) , (d) , (e) and (f). Solution 3.3.
3.4 Derivatives of Trigonometric Functions
Consider a function. means the sine of the angle whose radian measure is x. A similar convention holds for other trigonometric functions cos, tan, csc, sec and cot.
3.4.1 Derivatives and Example 3.4. Find if (i) (ii) (iii). Solution 3.4.
3.5 Implicit Differentiation
So far we have dealt with functions of the form. That is one variable explicitly expressed in terms of the other. Now consider which do not give neither variable say , in terms of the other, say. These functions are called implicit functions. The differentiation of such functions is known as implicit differentiation. In this method we simply treat as an unknown but differentiable function of and apply the rules of finding derivatives of , etc. Example 3.5. Find given. Solution 3.5. Note that if u is a function of x , then. That is. Example 3.5.
3.7 Partial Derivatives
3.7.1 Introduction Here we show one of the ways a function of several variables can be differentiated. Let. If we keep one of the variables, say , fixed then can be treated as a function of only and we can calculate the derivative (if it exists) of with respect to. This new function is called the partial derivative of with respect to and is denoted by. Example 3.7. Let. Find. Solution 3.7. Treating as if it were a constant, we have . Remark: The partial derivatives and give us the rate of change of as each of the variables and change with the other one held fixed. They do not tell us how changes when and change simultaneously. Although the functions and computed with one of the variables held constant, each is a function of both variables. Example 3.7. Let. Calculate and. Solution 3.7. . . The product and quotient rules for derivatives have counterparts for partial derivatives. Example 3.7.
Let. Find. Solution 3.7. . 3.7.2 Higher-Order Partial Derivatives A function of one variable may have second, third and higher derivatives. There is an analogue for functions of several variables. If is a function of the variables and , then the functions and may each have two partial derivatives. The partial derivatives of and would generate four new partial derivatives: . Example 3.7. Let. Find all second partial derivatives of. Solution 3.7. and. We obtain the second partials by computing the partial derivatives of the first partials:
