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Page 20 The Chain Rule is called the Power Rule, and recall that I said cant be done by the power rule because the base is an expression more complicated than x. In other words, in order to use the power rule, the base must be x, or the variable you are differentiating with respect to. However, it doesnt mean we cant differentiate (2x + 1) 3. All we need is a rule called the Chain Rule, more appropriately, Chain Rule with the Power Rule (three versions are provided, its up to you to choose the one you like): Here is how it works: If f(x) = 3(6 – 5x 2 ) 4, find f (x). Product Rule, Quotient Rule and Chain Rule (lets throw them together) Find f (x) for each of the following functions: 1. f(x) = (x – 2) 2 (x + 3)

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Page 21 Derivatives of Trigonometric FunctionsDerivative of sin x If f(x) = sin x, what is f (x)? Recall the limit definition of derivative: Chain Rule on sin (g(x)): If f(x) = sin (g(x)), then f (x) = cos (g(x)) g (x). That is, d / dx [sin (expression)] = cos (expression) d / dx (expression). Examples: For each of the following functions, find its derivative. 1. f(x) = sin x 2 f (x) = 2. f(t) = sin [(t + 2)(3t 2 – 4)] 3. g(x) =

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Page 22 Derivatives of Trigonometric FunctionsDerivative of cos x If f(x) =cos x, what is f (x)? This time we are not going to use the limit definition to find f (x), but rather, recall cos x = sin ( ): Chain Rule on cos (g(x)): If f(x) = cos (g(x)), then f (x) = ____________. That is, d / dx [cos (expression)] = _________________________. Examples: For each of the following functions, find its derivative. 1. f(x) = cos (x 2 + 2x – 1) f (x) = 2. f(t) = cos [(t – 3)(2t 2 + 1)] 3. g(x) =

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Page 23 Derivatives of Trigonometric FunctionsDerivatives of the Other Four If f(x) = tan x, what is f (x)? Recall: Chain Rule on these functions: If f(x) = tan (g(x)), then f (x) = ______________________If f(x) = cot (g(x)), then f (x) = ______________________ If f(x) = sec (g(x)), then f (x) = ______________________If f(x) = csc (g(x)), then f (x) = ______________________ If f(x) = cot x, what is f (x)? Recall: If f(x) = sec x, what is f (x)? Recall: If f(x) = csc x, what is f (x)? Recall: Examples: For each of the following functions, find its derivative. 1. f(x) = tan x 2 sin 2x f (x) = 2. f(t) = csc (t – 3) cot (2t 2 + 1) 3. g(x) =

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Page 24 Summary of Derivatives of All Six and Precaution on Chain Rules The table on the right shows the derivatives of the six basic trig. functions (notice the derivatives of the three cofunctionscosine, cotangent and cosecanthave a ________ sign). Of course, Chain Rule can be applied to each one of them (see bottom table). The order of applying Chain Rules We have to apply Chain Rule for finding the derivative of many functions, and for some of them, we need to apply Chain Rule more than once, and the ORDER we apply the Chain Rule MATTERS. Examples: Find the derivatives of the following functions. f(x) = sin x 3 vs. f(x) = sin 3 x f(x) = tan 2 (cos x) vs. f(x) = tan (cos 2 x) vs. f(x) = tan (cos x 2 ) f(x) = cot 2 [sin(x 2 + 3)]

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