121616949-math.93

# 121616949-math.93 - 4.7 EXAMPLE 4.6 Derivatives of the...

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4.7 Derivatives of the exponential and logarithmic functions 79 EXAMPLE 4.6 Compute the derivative of f ( x ) = 2 x 2 = 2 ( x 2 ) . d dx 2 x 2 = d dx e x 2 ln 2 = d dx x 2 ln 2 e x 2 ln 2 = (2 ln 2) xe x 2 ln 2 = (2 ln 2) x 2 x 2 EXAMPLE 4.7 Compute the derivative of f ( x ) = x x . At first this appears to be a new kind of function: it is not a constant power of x , and it does not seem to be an exponential function, since the base is not constant. But in fact it is no harder than the previous example.
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Unformatted text preview: d dx x x = d dx e x ln x = ± d dx x ln x ¶ e x ln x = ( x 1 x + ln x ) x x = (1 + ln x ) x x EXAMPLE 4.8 Recall that we have not justiﬁed the power rule except when the exponent is a positive or negative integer. We can use the exponential function to take care of other exponents. d dx x r = d dx e r ln x = ± d dx r ln x ¶ e r ln x = ( r 1 x ) x r = rx r-1...
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• Fall '07
• JonathanRogawski

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