The Derivative Of The Natural Logarithm - LTCC Online
| The Derivative of the Natural Logarithm Derivation of the Derivative Our next task is to determine what is the derivative of the natural logarithm. We begin with the inverse definition. If y = ln x then ey = x Now implicitly take the derivative of both sides with respect to x remembering to multiply by dy/dx on the left hand side since it is given in terms of y not x. ey dy/dx = 1 From the inverse definition, we can substitute x in for ey to get x dy/dx = 1 Finally, divide by x to get dy/dx = 1/x We have proven the following theorem
Examples Find the derivative of f(x) = ln(3x - 4) Solution We use the chain rule. We have (3x - 4)' = 3 and (ln u)' = 1/u Putting this together gives f '(x) = (3)(1/u) 3 = 3x - 4 Example find the derivative of f(x) = ln[(1 + x)(1 + x2)2(1 + x3)3 ] Solution The last thing that we want to do is to use the product rule and chain rule multiple times. Instead, we first simplify with properties of the natural logarithm. We have ln[(1 + x)(1 + x2)2(1 + x3)3 ] = ln(1 + x) + ln(1 + x2)2 + ln(1 + x3)3 = ln(1 + x) + 2 ln(1 + x2) + 3 ln(1 + x3) Now the derivative is not so daunting. We have use the chain rule to get 1 4x 9x2 f '(x) = + + 1 + x 1 + x2 1 + x3 Exponentials and With Other Bases
Examples Find the derivative of f (x) = 2x Solution We write 2x = ex ln 2 Now use the chain rule f '(x) = (ex ln 2)(ln 2) = 2x ln 2 Logs With Other Bases We define logarithms with other bases by the change of base formula.
Remark: The nice part of this formula is that the denominator is a constant. We do not have to use the quotient rule to find a derivative Examples Find the derivative of the following functions
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