Clicker Question 1 What is the slope of the tangent line to x y + x 3 = 4 at the point (1, 3)? A. 0 B. -3 C. -6 D. -10 E. (-3x 2 – y) / x.
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Clicker Question 1 What is the slope of the tangent line to x y + x 3 = 4 at the point (1, 3)? A. 0 B. -3 C. -6 D. -10 E. (-3x 2 – y) / x
Derivatives of Inverse Functions (3/18/09) The Chain Rule, or more specifically implicit differentiation, allows us to compute the derivatives of the inverses of any functions whose derivatives we already know. What is the inverse of x n (n whole)? What is the inverse of sin(x) (Etc.)? What is the inverse of a x ?
Derivatives of Root Functions Does the Power Rule extend to root functions? That is, if f (x ) = x 1/n, is f '(x ) = (1/n) x 1/n – 1 ? The answer is yes!, and we can see this by using implicit differentiation applied to both sides of the equation y n = x. Hence the Power Rule works for all positive and negative whole and fractional powers. (Check!)
Derivatives of Inverse Trig Functions We can use this same technique to discover the derivative formulas of inverse trig functions. Apply implicit differentiation to both sides of the equation sin(y ) = x to discover the derivative of the arcsin. Do the same for the arctan.
Clicker Question 2 What is the derivative of f (x ) = arcsin(x 2 ) ? A. 1 / (1 – x 2 ) B. 1 / (1 – x 4 ) C. arccos(x 2 ) D. 2x arccos(x 2 ) E. 2x / (1 – x 4 )
Derivatives of Log Functions Log functions are the inverses of exponential functions, and we know their derivatives. Hence we can use the procedure above to get their derivatives. The answer is: d(log a (x))/dx = 1 / (x ln(a)) In particular: d(ln(x))/dx = 1/x
Clicker Question 3 What is the instantaneous rate of change of g (x ) = x ln(x ) at x = e ? A. 1 / e B. 2 C. 1 / x D. 1 + e E. 1 + ln(x )
Now we have it all… With the addition of these facts about derivatives of inverse functions, we now have (almost) all the information we need (i.e., all the “rules” and “facts”) to find the derivative of any combination of standard functions given by formula. The only missing piece is a power function where the exponent is irrational.
Assignment for Friday On Page 214, do Exercises 45, 47, 49, 53, and 57. Read Section 3.6 to the middle of Page 217. On Page 220, do Exercises 3 – 23 odd, 31, 33, and 49.