find the derivative of the following Q: x²/2-5/x²
Answers
Answer:
We start with the derivative of a power function, f(x) = x
n
. Here n is a number of
any kind: integer, rational, positive, negative, even irrational, as in x
π
. We have already
computed some simple examples, so the formula should not be a complete surprise:
d
dxx
n = nxn−1
.
It is not easy to show this is true for any n. We will do some of the easier cases now, and
discuss the rest later.
The easiest, and most common, is the case that n is a positive integer. To compute
the derivative we need to compute the following limit:
d
dxx
n = lim
∆x→0
(x + ∆x)
n − x
n
∆x
.
For a specific, fairly small value of n, we could do this by straightforward algebra.
dxx
3 = lim
∆x→0
(x + ∆x)
3 − x
3
∆x
.
= lim
∆x→0
x
3 + 3x
2∆x + 3x∆x
2 + ∆x
3 − x
3
∆x
.
= lim
∆x→0
3x
2∆x + 3x∆x
2 + ∆x
3
∆x
.
= lim
∆x→0
3x
2 + 3x∆x + ∆x
2 = 3x
Answer:
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Explanation: