11. Differentiating inverse functions.

Transcription

1 . Differentiating inverse functions. The General Case: Fin y x when y f x. Since f y x an we regar y as being a function of x, we ifferentiate both sies of the equation f y x with respect to x using the chain rule for the left han sie. x f y y f y x y x. So, since y f x, x f x f f x Example. Fin x f x when f x 5 x 4 3. Here so f x y x f y, where y f x. So, while it is straightforwar to get the erivative as a function of y, we have to work harer to get it as a function of x : wehave, in fact, to fin the inversefunction: so x f x In fact it is mucheasier to get the erivative irectly whenwe have anexplicit expression for the inverse function. Here, x f x The chain rule metho of Example is useful when there is no simple formula for the inverse function as we shall see in the next section. Inverse trigonometric functions Earlier in this section we saw that one-to-one functions have inverse functions. But sin x is not a one-to-one function in general. It is not one-to-one for example when the omain is x. However, calculators have a key marke sin that is use to get one solution of the equation sin x a. They o this by restrictingthe omain of the function toan interval where it is one-to-one. Inverse sine function Below is the graph of sin x on the omain x an, alongsie it, the graph of sin x on the omain / x / x x x Domain,. Consier the function sin x, Domain, sin x, Domain, MA00: Calculus Chapter

2 f x sin x, x /, /. Since f x cos x is 0 for x /, /, it means that sin x is always increasing on this omain it is therefore one-to-one on this omain an must have an inverse (this is the largest omain containing the origin for which it is one to one). This means that the inverse function solves the equation sin x a for any value of a in the range, an gives the solution as a value of x which lies in the interval /, /. This is what calculators o. The inverse sine function is enote by sin x. also known as arcsin x an asin x. Function Domain sin x Range sin x If we trie any larger omain, sin x woul not be one to one an therefore woul not have an inverse. Below we show graphs of sin x, sin x an the line y x. The graph of sin x soli curve) is the reflection in the line y x of the graph of sin x (ashe curve) x The erivative of sin x We have to use the chain rule because there is no formula for sin x. Starting from y sin x we have x sin y an so, ifferentiating both sies, x x x sin y Now, because y is a function of x, weusethe chain rule to ifferentiatethe right han sie to get y x y x But cos y sin y x an, taking the square root (we take the positive square root because cosy is positive on the omain, ) we obtain cos y x So, finally we have x sin x x Inverse cosine function Below is the graph of cos x for x an alongsie it the graph of cos x plotte for 0 x x x x cosx, x,. cos x, x 0, cos x, x,. MA00: Calculus Chapter

3 Since cos x sin x is 0 for 0 x, it means that cosx is a ecreasing function on this interval an x is therefore one-to-one. This means that it has an inversein that interval the inversefunction solves the equation cos x a for any value of a in the interval, an gives the solution as a value of x which lies in the interval 0,. Function Domain Range cos x 0,, cos x, 0, The erivative of cos x We have to use the chain rule because there is no formula for cos x. Starting from y cos x we have x cos y an ifferentiating both sies gives x x x cos y. Now, using the chain rule to ifferentiate the right han, we get y x y x. But sin y cos y x so, taking the square root, sin y x (We take the positive square root because siny is positive on the omain 0,.) So, finally x cos x x. MA00: Calculus Chapter 3

4 A Curious Observation Notice that the erivative of sin x is x while the erivative of cos x is x : they are equal but oppositein sign. Why is this? To answer this, let y cos x so x cos y. However, we also have cos y sin y since sin y We therefore have sin y x from which we have y sin x, i.e., y We euce from this that y x x sin x x sin x but, from out original efinition, y cos x an so y x x cos x. A more geometrical view is to consier a right angle triangle with hypotenuse of unit length whose two smallest angles are an.we let x enote the length of the sie opposite. Then sin x an cos x while. Differentiating both sies of this equation gives 0, x i.e., sin x cos x 0. x Inverse tangent function The tangent function tan x is relate to the sine an cosine functions via tan x sin x cos x. The first graph shows tan x for x except that it is not possible to show it all because tan x has singularities (it tens to or ) at x /, / an 3 /. The secon shows a similar view of tan x for / x / x x x Since tanx,x,. tanx,x,. tan x, x R. tan x x we know that tan x is an increasing function for / x /. It is therefore a one to one function on the omain /, / an so must have an inverse. The inverse function tan x solves the equation tan x a for any value of a an returns the solution that lies in the interval /, /. Function Domain tan x, Range, tan x The erivative of tan x. Once more we have to use the chain rule because there is no formula for tan x. Starting from y tan x we have x tan y an so MA00: Calculus Chapter 4

5 x x x tan y. Now, using the chain rule to ifferentiate the right han sie, we have, since tan y y y x y x. Now use the trigonometric ientity sec y we fin an so y x x tan x x. Proof that sec y tan y. We start with sin y cos y an ivie both sies by cos y i.e. sin y cos y cos y sin y cos y an so tan y cos y sec y sec y. MA00: Calculus Chapter 5

6 Summary: Toprove theseresults weneee toknow. f f x x,. the chain rule, 3. the erivatives of sin y, cos y an tan y, 4. sin y cos y, 5. sec y tan y. Examples By further use of the chain rule we can show that x sin x x x cos x x x tan x x x sin x 4x x sin ax a a x x cos x 4x x cos ax x tan x x tan ax 4x a a x a a x For example, sin ax sin u, where u ax. So x sin ax u sin u u x u sin u x ax u a a ax. MA00: Calculus Chapter 6