How to derive cosh (3x) in terms of cosh (x) and sinh (x) ? Initially I start with 1/2 (e^3x+e^-3x) but I'm stuck from there because I don't know how to continue... :(
cosh(3x) = cosh(2x+x)
= cosh2x coshx + sinh2x sinhx
...
To derive cosh (3x) in terms of cosh (x) and sinh (x), the key is to use the exponential forms of cosh(x) and sinh(x):
cosh(x) = (e^x + e^(-x)) / 2
sinh(x) = (e^x - e^(-x)) / 2
Let's start with the expression you have:
cosh(3x) = (e^(3x) + e^(-3x)) / 2
To simplify this, let's focus on the numerator first. We can rewrite the numerator as a sum of two exponentials:
e^(3x) + e^(-3x) = e^(2x) * e^x + (e^(-2x) * e^x)
Now, notice that we can use the exponential forms of cosh(x) and sinh(x) to rewrite e^x and e^(-x):
e^(2x) * e^x + e^(-2x) * e^x = (cosh(2x) * (e^x) + sinh(2x) * (e^x))
Now, let's express cosh(2x) and sinh(2x) in terms of cosh(x) and sinh(x):
cosh(2x) = (e^(2x) + e^(-2x)) / 2
sinh(2x) = (e^(2x) - e^(-2x)) / 2
Substituting these back into our expression, we get:
(cosh(2x) * (e^x) + sinh(2x) * (e^x)) =
((e^(2x) + e^(-2x)) / 2) * (e^x) + ((e^(2x) - e^(-2x)) / 2) * (e^x)
Now, combine like terms:
((e^(2x) + e^(-2x)) / 2) * (e^x) + ((e^(2x) - e^(-2x)) / 2) * (e^x) =
((e^(3x) + e^(-x)) + (e^(3x) - e^(-x))) / 2
Now, simplify further:
((e^(3x) + e^(-x)) + (e^(3x) - e^(-x))) / 2 =
(2 * e^(3x)) / 2 = e^(3x)
Therefore, we have:
cosh(3x) = e^(3x)
So, the final result is cosh(3x) = e^(3x).
To derive cosh(3x) in terms of cosh(x) and sinh(x), you can use the identity cosh(2x) = cosh^2(x) + sinh^2(x). Here's how you can continue from where you left off:
1. Start with cosh(3x) = 1/2 (e^(3x) + e^(-3x)), which is correct.
2. Multiply and divide the expression by e^(x) to separate it into two parts:
cosh(3x) = 1/2 (e^(2x) * e^(x) + e^(2x) * e^(-x))
3. Notice that e^(2x) * e^(-x) = e^(x), so we can rewrite the expression:
cosh(3x) = 1/2 (e^(x) * e^(2x) + e^(x))
4. Now, let's use the identity cosh(2x) = cosh^2(x) + sinh^2(x):
cosh(2x) = cosh^2(x) + sinh^2(x)
5. Rearrange the terms:
sinh^2(x) = cosh(2x) - cosh^2(x)
6. Substitute sinh^2(x) in the expression for cosh(3x):
cosh(3x) = 1/2 (e^(x) * e^(2x) + cosh(2x) - cosh^2(x))
7. Use the identity cosh(x) = (e^x + e^(-x))/2:
cosh(3x) = 1/2 (e^(x) * e^(2x) + cosh(2x) - (e^x + e^(-x))/2)^2
8. Simplify the expression:
cosh(3x) = 1/2 (e^(3x) + e^(x) + e^(-x) + cosh(2x) - (cosh(2x))^2)
So, cosh(3x) can be expressed in terms of cosh(x) and sinh(x) as:
cosh(3x) = 1/2 (e^(3x) + e^(x) + e^(-x) + cosh(2x) - (cosh(2x))^2)
Keep in mind that this may not be the most simplified form, but it provides a way to express cosh(3x) using cosh(x) and sinh(x).
Using the chain rule:
u = 3 x
du = 3 dx Divide both sides by 3
du / 3 = dx
dx = du / 3
d cosh ( 3 x ) / dx =
d cosh ( u ) / ( du / 3 ) =
3 d cosh ( u ) / du
d cosh ( u ) / du = sinh u
so
d cosh ( 3 x ) / dx = 3 d cosh ( u ) / du = 3 sinh u = 3 sinh ( 3 x )