find the compute derivative of f(x)=sinh^2(5x)
d/dx sinh(u) = cosh(u) du/dx
f' = 2sinh(5x)cosh(5x)*5
= 10 sinh(5x)cosh(5x)
= 5sinh(10x)
To compute the derivative of the given function f(x) = sinh^2(5x), we can use the chain rule of differentiation. The chain rule states that if we have a composite function, f(g(x)), where f(u) and g(x) are differentiable functions, then the derivative of f(g(x)) is given by f'(g(x)) * g'(x).
In this case, let's break down the function f(x) = sinh^2(5x) into two parts: the inner function g(x) = 5x and the outer function f(u) = u^2, where u = sinh(5x).
First, let's calculate the derivative of the outer function, f(u) = u^2. The derivative of u^2 with respect to u is 2u. So, f'(u) = 2u.
Next, let's find the derivative of the inner function, g(x) = 5x. The derivative of 5x with respect to x is 5.
Now, using the chain rule, we can obtain the derivative of f(x) = sinh^2(5x) as follows:
f'(x) = f'(u) * g'(x)
= 2u * 5 (substituting u = sinh(5x) and g'(x) = 5)
= 10sinh(5x).
Therefore, the derivative of f(x) = sinh^2(5x) is 10sinh(5x).