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Lynna [10]
3 years ago
11

Find the derivative of f(x)= (e^ax)*(cos(bx)) using chain rule

Mathematics
1 answer:
Vikentia [17]3 years ago
4 0

If

f(x) = e^{ax}\cos(bx)

then by the product rule,

f'(x) = \left(e^{ax}\right)' \cos(bx) + e^{ax}\left(\cos(bx)\right)'

and by the chain rule,

f'(x) = e^{ax}(ax)'\cos(bx) - e^{ax}\sin(bx)(bx)'

which leaves us with

f'(x) = \boxed{ae^{ax}\cos(bx) - be^{ax}\sin(bx)}

Alternatively, if you exclusively want to use the chain rule, you can carry out logarithmic differentiation:

\ln(f(x)) = \ln(e^{ax}\cos(bx)} = \ln(e^{ax})+\ln(\cos(bx)) = ax + \ln(\cos(bx))

By the chain rule, differentiating both sides with respect to <em>x</em> gives

\dfrac{f'(x)}{f(x)} = a + \dfrac{(\cos(bx))'}{\cos(bx)} \\\\ \dfrac{f'(x)}{f(x)} = a - \dfrac{\sin(bx)(bx)'}{\cos(bx)} \\\\ \dfrac{f'(x)}{f(x)} = a-b\tan(bx)

Solve for <em>f'(x)</em> yields

f'(x) = e^{ax}\cos(bx) \left(a-b\tan(bx)\right) \\\\ f'(x) = e^{ax}\left(a\cos(bx)-b\sin(bx))

just as before.

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A building company bids on two large projects. The CEO believes the chance of winning the 1st is 0.6, the chance of winning the
Ksju [112]

Answer:

0.8

Step-by-step explanation:

A: P(1) = .6, P(2) = .5, P(1 and 2) = .3

P(1 or 2) = P(1) + P(2) - P(1 and 2) .6 + .5 - .3 = 0.8

4 0
3 years ago
Enter the value of p so that the expression 18n + 27 is equivalent to p(2n + 3)
Mamont248 [21]

Answer:

P = 9

Step-by-step explanation:

When P = 9, we could multiply 9 to (2n+ 3).

We would get:

18n + 27

---------------------------------------------------------------------------------------------------------------

Have a great summer :)

7 0
3 years ago
Negative solution for Multi-step equations
nydimaria [60]

Answer:

\frac{1}{2}(4x+6)+3x=7x+9

\frac{1}{2}(4x+8)+3x=7x+9

Step-by-step explanation:

By putting the numbers in the given equation,

\frac{1}{2}(4x+6)+3x=7x+9

2x + 3 + 3x = 7x + 9

5x + 3 = 7x + 9

(5x + 3) - 7x = (7x + 9) - 7x

(5x - 7x) + 3 = 9

-2x + 3 = 9

(-2x + 3) - 3 = 9 - 3

-2x = 6

-\frac{2x}{2}=\frac{6}{2}

-x = 3

x = -3

Therefore, our equation is correct which has a negative result.

In place of 6 if we use 8, value of x will be negative again.

3 0
3 years ago
Complete this word<br>[tex] _ r _ch_i_ [\tex]​
Anna007 [38]

Answer:

Arachnid is my guess.

3 0
2 years ago
A truck drive drove 48 miles in 45 minutes. At this rate, how many miles can the truck driver drive in one hour?
-Dominant- [34]

Answer:

the truck driver can drive 64 miles in one hour.

Step-by-step explanation:

Number of miles driven in 45 minutes = 48 miles

48/45x60=64

8 0
2 years ago
Read 2 more answers
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