If f(x)=4-coshx/3+coshx,
f'(x)=?
If you mean f(x)= (4-coshx)/(3+coshx),
Let u = 4 - coshx and v = 3 + coshx
u' = -sinhx v' = sinhx
f'(x) = (v u'- u v')/(v^2)
Finish it by substution
Got it. Thanks!
To find the derivative of the function f(x) = (4 - cosh(x))/(3 + cosh(x)), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative f'(x) can be found using the formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x))/[h(x)]^2
Let's find the derivatives of g(x) and h(x) first:
g(x) = 4 - cosh(x)
The derivative of cosh(x) is sinh(x), so the derivative of g(x) with respect to x is:
g'(x) = 0 - sinh(x) = -sinh(x)
h(x) = 3 + cosh(x)
The derivative of cosh(x) is sinh(x), so the derivative of h(x) with respect to x is:
h'(x) = 0 + sinh(x) = sinh(x)
Now that we have the derivatives of g(x) and h(x), we can use the quotient rule to find the derivative of f(x):
f'(x) = (g'(x) * h(x) - g(x) * h'(x))/[h(x)]^2
Substituting the values, we have:
f'(x) = (-sinh(x) * (3 + cosh(x)) - (4 - cosh(x)) * sinh(x))/[(3 + cosh(x))^2]
Simplifying this expression gives us the derivative:
f'(x) = [-sinh(x) * 3 - sinh(x) * cosh(x) + sinh(x) * 4 - sinh(x) * cosh(x)]/[(3 + cosh(x))^2]
Combining like terms:
f'(x) = -6sinh(x)/(3 + cosh(x))^2
Therefore, the derivative of f(x) with respect to x is f'(x) = -6sinh(x)/(3 + cosh(x))^2.