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1 MATH 1170 Section 3.1 Worksheet NAME Recall that we have efine the erivative of f to be f (x) = lim h 0 f(x + h) f(x) h Recall also that the erivative of a function, f (x), is the slope f s tangent line at the point (x, f(x)). We are now going to explore what generalizations we can make about the certain kin of functions we have previously iscusse. Let s begin with polynomial functions. The most basic polynomial function is the constant function. Constant Functions Recall that a constant function is a function f(x) = c where c is a constant (real number). Sketch a graph of the function f(x) = c. What is the slope of this line? Looking at the graph, what woul you imagine that the slope a line tangent to f(x) = c at any given is? Using the efinition of erivative, compute the erivative of f(x) = 3. Using the efinition if erivative, compute the erivative of f(x) = c. 1

2 Derivative of the Constant Function x (c) = This little rule that we establishe makes it easy to compute erivatives of functions like: f(x) = 90 g(t) = 17, 562 h(r) = π 2 /4 k(x) = e 1000 s(t) = 2...they are all...0! Power Functions Sketch a graph of f(x) = x. Using the efinition of the erivative, compute the erivative of f(x) = x. Using the efinition of erivative, compute the erivative of f(x) = x 2. Using the efinition of erivative, compute the erivative of f(x) = x 3. 2

3 Using the efinition if erivative, compute the erivative of f(x) = x n for where n > 0 is an integer. Hint: Recall the Binomial Theorem states that (x + h) n = a n,0 x n + a n,1 x n 1 h + a n,2 x n 2 h a n,n 2 x 2 h n 2 + a n,n 1 xh n 1 + a n,n h n where a i,j is the j th entry in the i th column of Pascal s Triangle. This means that (x + h) n = x n + nx n 1 h + a n,2 x n 2 h a n,n 2 x 2 h n 2 + nxh n 1 + h n. We have now establishe x (xn ) where n is a positive integer. While it is a little more complicate to prove this result when n is not an integer, what you showe above oes exten for n any real number. If n is any real number, The Power Rule x (xn ) = Practice Fin the erivative of the following functions using the Constant Rule or Power Rule. f(x) = x 5 g(t) = t 4 h(x) = π 2 h(t) = t 1000 f(r) = r 5/4 g(x) = x k(x) = x π s(r) = r 0 h(x) = 3 x 2 3

4 Exponential Functions Recall that an exponential function comes in the form of f(x) = a x where a is some constant. Note that these functions are very ifferent from power functions that have the form f(x) = x a where a is a constant. Sketch the graphs of f(x) = 2 x an f(x) = x 2. Using the efinition of erivative, compute the erivative of f(x) = a x. Compare lim h 0 a k 1 h to the efinition of the erivative of f(x) = a x at 0. What o you notice? The Derivative of the Exponential Function x (ax ) = The efinition of e is such that e h 1 lim = 1. h 0 h What oes this tell you about the erivative of e x? 4

5 The Derivative of the Natural Exponential Function x (ex ) = Composition of Derivatives Now that we have establishe a few basic rules for erivatives, it woul be beneficial to figure out how to combine them. For instance, we know that the erivative of f(x) = x 5, But what about g(x) = 2x 5? x (x5 ) = 5x 5 1 = 5x 4. Using the efinition of erivative, compute the erivative of f(x) = 2x.( ) Using the efinition if erivative, compute the erivative of f(x) = 3x 2.( ) Using the efinition of erivative, compute the erivative of g(x) = cf(x) where c is a constant an f is a ifferentiable function. Hint: Refer to the rules of limits on the review sheet. 5

6 The Constant Multiple Rule If c is a constant an f is a ifferentiable function, then x (cf(x)) = Verify that the ( ) erivatives can be compute (resulting in the same solution) using the Constant Multiple Rule. Practice Using the Constant Multiple Rule (an the other erive rules), compute the following erivatives: g(x) = 2x 5 f(t) = 3 t h(x) = 1 3 ex k(x) = ex 4 f(x) = 5π 2 g(t) = 5x 2/3 4 We are clearly making progress, but what about more complicate polynomials like f(x) = x 2 2x + 1? Using the efinition of erivative, compute the erivative of F (x) = f(x) + g(x). Hint: You may want to refer to the rules of limits on the review sheet. 6

7 The Sum Rule If f an g are ifferentiable functions, then (f(x) + g(x)) = x Using the Sum Rule, compute the erivative of f(x) = x 2 + x. Can you make a conjecture about the erivative a the ifference of two ifferentiable functions? What o you think that x (f(x) g(x)) will be an why? The Difference Rule If f an g are ifferentiable functions, then (f(x) g(x)) = x Putting it Together Differentiate the following functions using the rules that you establishe. a. f(x) = x e x b. f(x) = 3x 4 ( x ) c. f(x) = 4x 2 2x x 1. f(x) = 3 x e. f(x) = 3x 5 + e x π f. f(x) = x 1/2 (x 3 + 1/x) 7

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