Derivative of inverse trigonometric functions pdf

View Derivative of Inverse Trigonometric Functions.pdf from MIL 125 at Lyceum of the Philippines University - Cavite - General Trias, Cavite. Derivatives of Inverse 7.4 derivatives of inverse trigonometric functions 539 7.4 Derivatives of Inverse Trigonometric Functions The three previous sections introduced the ideas of one-to-one func-tions and inverse functions, then used those concepts to define arcsine, arctangent and the other inverse trigonometric functions. In this section, we obtain derivative We can use the formulas for the derivatives of the trigonometric functions to prove formulas for the derivatives of the inverse trigonometric functions. Interestingly, although inverse trigonometric functions are transcendental, their derivatives are algebraic: Theorem 2.18 Derivatives of Inverse Trigonometric Functions 5) Use logarithmic di erentiation to nd the derivative of each of the following functions: (a) y= xsinx(b) y=x2 3 p 5+2 (x+2)5 6) (a) We proved the power rule (xn)0= nxn 1for the case when nwas a positive integer and in some other special cases. Derivatives, Integrals, and Properties Of Inverse Trigonometric Functions and Hyperbolic Functions (On this handout, a represents a constant, u and x represent variable quantities) De rivatives of Inverse Trigonometric Functions d dx sin¡1u = 1 p 1¡u2 du dx (juj < 1) d dx cos¡1u = ¡1 p 1¡u2 du dx (juj < 1) d dx tan¡1u = 1 1+u2 This is an application of the chain rule together with our knowledge of the derivative of ex. d dx (e3x2)= deu dx where u =3x2 = deu du × du dx by the chain rule = eu × du dx = e3x2 × d dx (3x2) =6xe3x2. Example Find d dx (e x3+2). Solution Again, we use our knowledge of the derivative of ex together with the chain rule. d dx (ex3+2x)= deu. Derivatives of Inverse Trig Functions y = arcsin x y = arccos x y = arctan x y = arccot x y = arcsec x y = arccsc x These can be written as y = sin-1x rather than y = arcsinx sin-1x does NOT mean 1 sinx. 5 Example 3: Evaluate the derivative of sin y = x. 6 Introduction Trigonometric Functions (With Restricted Domains) and Their Inverses The Inverse Cosine Function The Inverse Tangent Function Definition of the Inverse Cotangent Function Formula for the Derivative of Inverse Secant Function Formula for the Derivative of Inverse Cosecant Function Important Sets of Results and their Applications Use implicit di erentiation to nd derivatives of inverse trigonometric functions. Review of Inverse Functions: To nd the inverse of a function, reverse the roles of input and output. If f(a) = bthen f1(b) = a . If f(a) = f(c) = bfor some a6=c, we have a problem|what should f1(b) be? Derivatives Involving Inverse Trigonometric Functions y x 5, y t 0 17 14 6) Find the derivative of the inverse For example, the sine function (x Free worksheet(pdf) and answer key on Inverse Functions--identify, write and express the inverse of functions based on graphs, tables, order pairs and more Free worksheet(pdf) and answer key on Inverse Functions--identify, write and express the Derivative of the Inverse Cosine Function d dx (cos1x) = 2 The Inverse Tangent Function Use simliar techniques as above to nd the derivative of y= tan1x. Note: You may need to refer to your other trigonometric identities. Derivative of the Inverse Tangent Function d dx (tan1x) = 3 Practice Exercises Derivative of the Inverse Cosine Function d dx (cos1x) = 2 T