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Unformatted text preview: Math 1a Derivatives & Logarithms Fall, 2009 1 In this problem, we’ll figure out the derivative of ln( x ) and log a ( x ) (where a is a positive constant other than 1). We’ll do this in the same way we found the derivatives of arcsin( x ), arccos( x ), and arctan( x ) on Monday, as once again we’re dealing with an inverse function. (For example, arcsin( x ) is the inverse function of sin( x ): sin(arcsin( x )) = x for any x , and sin(arcsin( x )) = x for x in the interval [- π 2 , π 2 ].) (a) Write the equation y = ln( x ) in terms of only x , y , and e . (b) Differentiate the equation you found in part (a) implicitly and solve for dy dx . (c) Now write dy dx only in terms of x (and not y ). This is the derivative of ln( x ). (d) Repeat this process for the function y = log a ( x ), where a 6 = 1 is a positive constant. 2 Using the chain rule, we can expand the derivative rules you found to d dx (ln( u )) = 1 u du dx or d dx (ln( g ( x ))) = g ( x ) g ( x ) and d dx (log a ( u )) = 1 ln( a ) u du dx or d dx (log a ( g ( x ))) = g ( x ) ln( a ) g ( x ) Use these rules to differentiate the following: (a) ln(cos( x )) (b) ln( √ x )...
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- Fall '09
- Derivative, Logarithm, clare, dx ln