Prove the Leibniz rule for
f
(
n
)
(
x
)
,
where
f
(
n
)
is the
n
th derivative of
f
;
that is, show that
(
f
g
)
(
n
)
(
x
)
=
n
∑
k
=
0

(
n
k
)
f
(
k
)
(
x
)
g
(
n
−
k
)
(
x
)
.

Check Explanation

Step-by-step explanation:

We want to prove that

(fg)ⁿ(x) = Σ ⁿCₖ fᵏ(x) gⁿ⁻ᵏ(x) (summing from k=0 to k = n)

Note that

(fg)ⁿ is the nth derivative of the product of f(x) × g(x)

ⁿCₖ is n combination k

fᵏ(x) is the kth derivative of f(x)

gⁿ⁻ᵏ(x) is the (n-k)th derivative of g(x)

To prove,

Let f(x) = u and g(x) = v

And f'(x) = u', g'(x) = v', etc.

We start with when n = 1

(uv)'

Using the product rule for derivatives,

(uv)' = (u'v + uv')

For n = 2

(uv)" = [(uv)']' = (u'v + uv')' (we use the product rule on each combination and the addition rule to sum up all the derivatives)

(u'v + uv')' = u"v + u'v' + u'v' + uv"

= u"v + 2u'v' + uv"

n = 3

(uv)''' = [(uv)"]' = [u"v + 2u'v' + uv"]'

= u'''v + u"v' + 2u"v' + 2u'v" + u'v" + uv'''

= u'''v + 3u"v' + 3u'v" + uv'''

n = 4

(uv)"" = [(uv)''']' = [u'''v + 3u"v' + 3u'v" + uv''']'

= u""v + u'''v' + 3u'''v' + 3u"v" + 3u'v''' + u'v''' + uv""

= u""v + 4u'''v' + 6u"v" + 4u'v''' + uv"" + uv""

At tbis stage, it is evident that the coefficients of each term in the final expression for the nth derivative of the product of two functions matches the Pascal's triangle equivalent.

Furthermore, it is clear that these formulas are similar to the binomial expansion raised to the appropriate exponent. With the terms with zero exponent u⁰ and v⁰ corresponding to the functions u and v themselves.

We can then write the general formula for the nth derivative of the product of two functions, uv, using binomial theorem's general formula as

(uv)ⁿ = Σ ⁿCₖ uᵏ vⁿ⁻ᵏ (summing from k = 0 to k = n)

(uv)ⁿ isn't the nth power of (uv) like the normal binomial expansion, instead, it is the nth derivative of the product of u and v.

ⁿCₖ is still n combination k

uᵏ isn't the kth power of u like in the normal binomial expansion, instead, it is the kth derivative of u

vⁿ⁻ᵏ isn't the (n-k)th power of v, instead, it is the (n-k)th derivative of v

We can then substitute back u = f(x) and v = g(x)

(fg)ⁿ(x) = Σ ⁿCₖ fᵏ(x) gⁿ⁻ᵏ(x) (summing from k=0 to k = n)

(Proved!)

Hope this Helps!!!

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

the final amount you will pay is $68.04.

the tip is determined by the pre-tax amount, so we calculate the tip based on the original bill, not after we've added sales tax.

to find the tip:

we can divide 54.00 by 100 to get 1% of 54. then we multiply that by 18.

54 / 100 = .54

.54 x 18 = 9.72

tip = 9.72$

to find the tax:

we can divide 54.00 by 100 to get 1% of 54. then we can multiply that by 8 to get 8%.

54 / 100 = .54

.54 x 8 = 4.32

tax= 4.32

now we can add the tip and tax to the bill to get our total.

54 + 9.72 + 4.32 = 68.04

total: $68.04

step-by-step explanation: