Lecture 9 4.1: Derivative Rules MTH 124

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1 Today we will see that the derivatives of classes of functions behave in similar ways. This is nice because by noticing this general pattern we can develop derivative rules which will make taking derivative quick and 1 painless. We can convince ourselves why these rules work intuitively using graphs and tables. We can convince ourselves why these rules work rigorously by using the definition of the derivative. For example consider a constant function f(x) = c, where c is just some constant(so c could be 3 or 78.4 for example). 1. Apply the definition of the derivative to f(x) = c, where c is a constant, to determine f (x). Since c could be any constant we ve just determined a derivative rule given any derivative function. Derivative of constant functions The derivative of any function of the form f(x) = c, where c is some constant is given by f (x) =. Recall in the previous lecture we determined the derivative of a general linear function of the form f(x) = mx + b where m and b are some constants. Put another way, given any linear function we can determine its derivative according to this rule. Derivative of linear functions The derivative of a function of the form f(x) = mx + b, where m and b are constants, is given by f (x) =. 1 Hopefully! 49

2 Generalizing Rules By taking the derivative of constant functions and linear functions we were actually taking the derivative of powers of x. We can generalize this more with a rule that allows us to take the derivative of any power of x. This rule is known as the power rule and is stated below. The Power Rule The derivative of the general power function f(x) = x n, where n is any real number, is given by (x n ) = nx n 1. Example 1 Determine the derivative of g(x) = x 5. First notice our function is a power function so this indicates the power rule is the appropriate choice. Applying the power rule we have g (x) = 5x 5 1 = 5x 4, so we see g (x) = 5x Use the power rule to determine the derivative of P (t) = t 0.2. Two other important and useful rules are given below. Derivatives of Sums, Differences, and Constant Multiples If f(x) and g(x) are two differentiable functions, and c some real number, then the sum, f(x) + g(x), the difference, f(x) g(x), and the constant multiple, cf(x), are differentiable, and [f(x) ± g(x)] = f (x) ± g (x) [cf(x)] = cf (x) We now consider a few examples to see how these rules are applied to problems. 50

3 Example 2 Determine the derivative of k(x) = x 2 + 3x. We can think of k(x) as the sum of two functions f(x) = x 2 and g(x) = 3x. In other words, using the rule about sums of functions we have that ( k(x) ) = ( f(x) + g(x) ) = ( x 2 + 3x ) = ( x 2) + ( 3x ) (sum rule) Now we ve reduced our problem down to two derivatives. The first is a power function so we can apply the power rule. The second term is a power function with a constant multiple so we ll use both the power rule and constant multiple rule. This gives us k (x) = ( x 2) ( ) + 3x = 2x (x ) (power and constant multiple rule) = 2x + 3(1x 1 1 ) (power rule) = 2x + 3x 0 = 2x + 3(1). So from our work above we see that k (x) = 2x Practice using the rules by finding the derivatives of the following functions. (a) A(t) = 3t 5 (b) p(x) = 12x 1.5 (c) q(x) = x 4 2x

4 Example 3 The annual net income of General Electric for the period of could be approximated by where t is years since P (t) = 3t 2 24t + 59 billion dollars (2 t 6), (a) How fast was GE s annual net income changing in 2010? (b)according to the model, GE s annual net income (A) increased at a faster and faster rate. (B) increased at a slower and slower rate. (C) decreased at a faster and faster rate. (D) decreased at a slower and slower rate. (E) none of the above. 3. The median home price in the United States over the period can be approximated by P (t) = 5t t 30 dollars (3 t 11), where t is time in years since the start of (a) Determine P (t) and P (6). (b) What does your answer for P (6) tell you about home prices? Considering the domain of P (t), what does P (t) tell you about home prices? 52

5 4. Use derivative rules to determine the derivatives of the following functions. (a) q(t) = ( t) 3 (b) g(t) = t t (c) f(x) = 1 x (d) r(x) = 34x x 3 (e) g(t) = 5t2 7t 3 t (f) h(x) = 3 x 2 + x x 4 53

6 5. The cost in dollars of producing q units of a product is given by C(q) = 250q + q 2. Find C(100) and C (100), give units and interpret your answers. 6. For this problem let f(x) = x 3 2x 2 + 6x 3. (a) Sketch f(x) and draw the tangent line at x = 1. (b) Find the equation for the tangent line of f(x) at x = Recall the equation of the tangent line of f(x) at the point x = 1 is the line y = mx + b that goes through the point (1, f(1)) whose slope is f (1). 54