9514 1404 393
Answer:
1. f'(x) = 0
3. f'(x) = -2x^-3
9. f'(x) = 10/9∛x
12. 2/3x^(-1/3) +5 +5x^7
Step-by-step explanation:
The two basic rules being used here are ...
d(au)/dx = a(du/dx) . . . . . multiplication by a constant
d(u^n)/dx = n·u^(n-1)·du/dx . . . . . power rule
In each case, u can be any function of x.
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When radicals are involved, it is often useful to write the root as an exponent. Then application of the power rule is more obvious.
In problem 14, it is convenient to do the multiplication before the differentiation. Otherwise the product rule gets involved, and the answer is a bit messy before being simplified.
Since these all use the power rule, we have shown a few examples. Where fractional or negative exponents are involved, you can simplify in the usual way, and write using radicals as you may choose. Here, we're really only concerned with a proper expression of the derivative.
1. f'(x) = 0
3. f'(x) = -2x^-3 . . . . . . . . . compare to problem 7
9. f'(x) = 10/9∛x
12. 2/3x^(-1/3) +5 +5x^7
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Additional comment
Another way to write the power rule is ...
d(x^n)/dx = (n·x^n)/x
This can be useful when n is a fraction or less than 1. For example, in problem 12, it keeps the radical out of the denominator, effectively eliminating a simplification step.
