We want to find the derivative of y(x) = ln(cosh(x) + sqrt(cosh^2(x) - 1)) with respect to x. To do this, we'll use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. So:
D[y(x)] = D[ln(u)] * D[u]
Where u = cosh(x) + sqrt(cosh^2(x) - 1).
First, let's find the derivative of the outer function, ln(u). The derivative of the natural logarithm is 1/u:
D[ln(u)] = 1/u.
Now, let's find the derivative of the inner function, u = cosh(x) + sqrt(cosh^2(x) - 1). We'll need to use the chain rule again, as well as the derivative of the hyperbolic cosine function, which is the hyperbolic sine function (sinh(x)):
D[u] = D[cosh(x)] + D[sqrt(cosh^2(x) - 1)]
D[u] = sinh(x) + (1/2)(cosh^2(x) - 1)^(-1/2) * D[cosh^2(x) - 1]
D[u] = sinh(x) + (1/2)(cosh^2(x) - 1)^(-1/2) * (2*cosh(x)*sinh(x) - 0)
D[u] = sinh(x) + (cosh(x)*sinh(x))/(sqrt(cosh^2(x) - 1))
Now that we have both the derivatives of the outer and inner functions, we can apply the chain rule:
D[y(x)] = D[ln(u)] * D[u] = (1/u) * D[u].
Substituting the expressions for the inner function (u) and its derivative (D[u]):
D[y(x)] = 1/(cosh(x) + sqrt(cosh^2(x) - 1)) * (sinh(x) + (cosh(x)*sinh(x))/(sqrt(cosh^2(x) - 1)))
This is the derivative of y(x) with respect to x.
Let y(x)=ln(cosh(x)+ sqrt(cosh^2(x)−1))
Find the derivative Dxy=
I don't know what to do with this...
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