Let y(x)=ln(cosh(x)+ sqrt(cosh^2(x)−1))
Find the derivative Dxy=
I don't know what to do with this...
To find the derivative of the function y(x) = ln(cosh(x) + sqrt(cosh^2(x) - 1)), we can use the chain rule and derivative rules for logarithmic functions.
Let's go step by step:
Step 1: Identify the composition of functions
In the given function, we have a composition of two functions: the natural logarithm function (ln) and the expression inside, which involves the hyperbolic cosine function (cosh) and a square root (√).
Step 2: Find the derivative of the inner function
To differentiate the inner function, we use the chain rule. The derivative of cosh(x) with respect to x is sinh(x), and the derivative of the square root of a function u(x) is 1/(2√u(x)).
So, the derivative of the inner function is (sinh(x) + 1/2 √(cosh^2(x) - 1)).
Step 3: Apply the derivative of logarithmic functions
The derivative of ln(u) with respect to x is 1/u multiplied by the derivative of u with respect to x. In this case, u(x) is the inner function, so its derivative is (sinh(x) + 1/2 √(cosh^2(x) - 1)).
Thus, the derivative of ln(cosh(x) + sqrt(cosh^2(x) - 1)) with respect to x is 1/(cosh(x) + sqrt(cosh^2(x) - 1)) multiplied by (sinh(x) + 1/2 √(cosh^2(x) - 1)).
To summarize, the derivative Dxy of y(x) is:
Dxy = 1/(cosh(x) + sqrt(cosh^2(x) - 1)) * (sinh(x) + 1/2 √(cosh^2(x) - 1))