Using exponents law indiscriminately in Calculus I
- From: CB4ever <CB4ever@xxxxxxxxxxxxxxx>
- Date: Sun, 08 Jan 2006 22:28:32 -0500
The exponent laws are defined for positive bases, yet in Calculus I much manipulation using these laws are done on expressions that aren't guaranteed to be positive. Can someone with experience in Calc 1 (especially the section on chain rule) and it's many manipulations of expressions using exponent laws tell me why we are safe in doing this?
For example
f(x) = x arctan[sqrt_x]
f'(x) = arctan[sqrt(x)] + x * 1/[1 + (sqrt_x)^2] * 1/(2 sqrt_x)
Is simplified (using exponent laws) to
f'(x) = arctan[sqrt_x] + sqrt_x /( 2 + 2x)
What makes it safe for us to do this? Application of the chain rule gave me the above messy f'(x) expression (which is not defined at 0) , but after simplification I get an expression that is defined at 0.
This is a simple example, there are many cases (especially in the section on chain rule) where exponent laws are used on large expressions (not just x as in my example). What makes this safe?
Forward thank you for help. .
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