Calculus 220 section 1 6 Some Basic Properties of Derivatives notes by Tim Pilachowski Today we take the ideas of section 1 3 the first derivative and expand a little bit Given a function y k f x where k is a constant coefficient of a variable function what would be its first derivative Think in terms of what you know about transformations and what you ve learned about slope of the tangent line The constant will either stretch the graph when k 1 or shrink the graph when k 1 What effect will this stretch shrink have on the slope of the tangent line y x 2 y 2x 2 Will it stretch shrink at the same rate as the curve Consider the quadratics pictured to the left For f x x 2 f 1 1 and the slope of the tangent line 2 For f x 2 x 2 f 1 2 and the slope of the tangent line 4 We might begin to suspect that as a function undergoes a stretch shrink the tangent line stretches shrinks at the same rate Indeed if we were to do more with limits it would be easy to show that kf x h kf x k f x h f x f x h f x lim lim k lim h 0 h 0 h 0 h h h dy k f x constant multiple rule which is to say that given a function y k f x dx Example A Given f x 7x 3 find the derivative Answer 21x2 Can we do the same if we add two functions together That is Given f x h g x h f x g x p x f x g x does p x f x g x Since lim h 0 h f x h f x g x h g x lim f x h f x lim g x h g x the answer is Yes lim h 0 h 0 h 0 h h h The derivative of a sum is the sum of the derivative sum rule Note that the constant multiple rule and sum rule work together to give us the derivative of a subtraction since f x g x f x 1 g x Example B Given f x x 3 8 x 2 find the first derivative Answer 3x 2 8 Example C Given g x 2 x 5 x4 7 1 14 3 3 x 2 find the first derivative Answer 10 x 4 x 3 2 3 4 x x 3 x Caution Be careful Warning Warning Danger Will Robinson There is no similar easy process for the derivative of a product nor is there a similar easy process for the derivative of a quotient We ll need to work a good bit for those Until we derive them however there are still a few things we can do Example D Given m x 4 x 2 1 3 x 3 find m Answer 20 x 4 3x 2 24 x Example E Given n x 5 x 4 1 find n x Answer 200 x 7 40 x 3 2 One more property rule to discuss Your text calls it the general power rule It s one version of something that later on we ll call the chain rule which applies to differentiating a composition of functions d g x r r g x r 1 d g x dx dx Although it s not a rigorous mathematical description I tend to think of this as the derivative of the outside applied to the inside times the derivative of the inside Example E revisited Given n x 5 x 4 1 find n x Answer 40 x 3 5 x 4 1 2 9 2x 1 dy Example F one more for the fish Given y 3x 2 2 find dx x 2x 1 Answer 9 3x 2 x2 8 2 2 6 x x 2 x3 Note the clever way we manipulated the quotient into a sum of power functions being careful to distribute the negative and using exponent properties to simplify
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