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Unformatted text preview: 3.7 Derivatives of Inverse Functions
Derivatives of Inverses of Differential Functions
What is the inverse function of f (x) =
x 2 + 1? 1 THEOREM 4  The Derivative Rule for Inverses If f has an interval I as domain and f (x) exists and is never zero on I, then f 1 is differentiable at every point in its domain. The value of (f 1 ) at a point b in the domain of f 1 is the reciprocal of the value of f at the point a = f 1 (b): Example 1 f (x) = x2 , find (f 1 ) (x) ( derivative of the inverse ) 2 Derivative of the Natural Logarithmic Function
Example Solve for x: 17 = 19x What is the inverse of f (x) = ex ? (f 1 ) (x) = d (ln x) = dx If u is a differentiable function of x with u > 0, we get ( applying the Chain Rule ) d (ln u) = dx Example 3 1.
d dx ln 2x 3 2. d dx ln(x2 + 3) The Derivatives of au and loga (u)
ax = d x a = dx If a > 0 and u is a differentiable function of x, then au is a differentiable function of x and d u a = dx loga (x) =
ln x ln a For a > 0 and a = 1, d loga u = dx Example g(t) =
3 log (sec(t)) 4 Logarithmic Differentiation
Example 4 Find y if (x2 + 1)(x + 3) 2 , y= x1
1 x>1 Example y = xx
2 5 Line Equation
If a line y goes through the point (a, b) and has slope m, then the equation for y is This is very useful when you have to find tangent and normal lines! Example m = 1 and the line goes through (1, 1). 6 ...
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 Spring '08
 FBHinkelmann
 Calculus, Derivative, Inverse Functions

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