Find the derivatives of the function f(y)= sinh(sinh(7y).
recall d(sinh(x)/dx = cosh(x)
so if
y = sinh(sinh(x))
dy/dx = cosh(sinh(x) * cosh(x)
= cosh(x) (cosh(sinh(x) )
so f ' (y) = 7 cosh(7y) (cosh(sinh(7y) )
thank you
To find the derivative of the function f(y) = sinh(sinh(7y)), we will use the chain rule.
The chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by the product of the derivative of f with respect to g and the derivative of g with respect to x. In other words:
d/dx[f(g(x))] = f'(g(x)) * g'(x)
Let's proceed with finding the derivative of f(y) step by step:
Step 1: Let's start by finding the derivative of the inner function g(y) = sinh(7y).
The derivative of sinh(u) with respect to u is given by sinh'(u) = cosh(u). Therefore, the derivative of g with respect to y is:
g'(y) = cosh(7y)
Step 2: Now, let's find the derivative of the outer function f(y) = sinh(g(y)).
The derivative of sinh(u) with respect to u is given by sinh'(u) = cosh(u). Therefore, the derivative of f with respect to g is:
f'(g) = cosh(g)
Step 3: Finally, using the chain rule, we can find the derivative of f(y) as follows:
f'(y) = f'(g) * g'(y)
Substituting the derivatives we found in steps 2 and 1:
f'(y) = cosh(g) * cosh(7y)
Since g(y) = sinh(7y), we can rewrite this as:
f'(y) = cosh(sinh(7y)) * cosh(7y)
Therefore, the derivative of f(y) = sinh(sinh(7y)) is cosh(sinh(7y)) * cosh(7y).