Find the derivatives of the function f(t)=cosh^2(4t)-sinh^2(4t). Simplify your answer completely.
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do you recall that cosh^2 - sinh^2 = 1?
To find the derivative of the function f(t) = cosh²(4t) - sinh²(4t), we need to use the chain rule and the derivative rules for hyperbolic functions.
Let's start by finding the derivative of the first term, cosh²(4t):
f'(t) = (cosh²(4t))'
To find the derivative of cosh²(4t), we can rewrite it as cosh(4t) * cosh(4t):
f'(t) = (cosh(4t) * cosh(4t))'
Now, using the chain rule, we differentiate each factor separately:
f'(t) = (cosh(4t))' * cosh(4t) + cosh(4t) * (cosh(4t))'
The derivative of cosh(4t) is sinh(4t) multiplied by the derivative of the inner function (4t)'= 4:
f'(t) = sinh(4t) * cosh(4t) + cosh(4t) * (4 * t)'
Next, let's simplify the second term, sinh²(4t):
f'(t) = sinh²(4t)'
Using the chain rule again, we differentiate sinh²(4t) as (sinh(4t))' * sinh(4t) + sinh(4t) * (sinh(4t))':
f'(t) = (sinh(4t) * cosh(4t)) * sinh(4t) + sinh(4t) * (sinh(4t))'
The derivative of sinh(4t) is cosh(4t) multiplied by the derivative of the inner function (4t)'= 4:
f'(t) = (sinh(4t) * cosh(4t)) * sinh(4t) + sinh(4t) * (4 * t)'
Now, let's simplify our expression:
f'(t) = sinh²(4t) * cosh(4t) + sinh(4t) * 4 * t
Finally, we can factor out a 4t from the second term:
f'(t) = sinh²(4t) * cosh(4t) + 4t * sinh(4t)
This is the simplified expression for the derivative of f(t).
To find the derivatives of the given function f(t) = cosh^2(4t) - sinh^2(4t), we will apply the chain rule and derivative rules for hyperbolic functions.
Step 1: Find the derivative of cosh^2(4t).
The derivative of cosh^2(4t) can be found by using the chain rule. The chain rule states that if we have a composite function f(g(t)), then its derivative is f'(g(t)) * g'(t).
In this case, f(t) = cosh^2(t), so we have to find the derivative of the outer function first (f') and then multiply it by the derivative of the inner function (g').
The derivative of cosh(t) is sinh(t), so:
f'(t) = 2 * cosh(4t) * sinh(4t)
Step 2: Find the derivative of sinh^2(4t).
Using a similar approach, we differentiate the sinh^2(4t) term.
The derivative of sinh^2(t) is 2 * sinh(t) * cosh(t), so:
g'(t) = 2 * sinh(4t) * cosh(4t)
Step 3: Combine the results from steps 1 and 2.
Now that we have f'(t) = 2 * cosh(4t) * sinh(4t) and g'(t) = 2 * sinh(4t) * cosh(4t), we can substitute these values back into our original function to get the complete derivative of f(t).
f'(t) = f'(t) - g'(t)
= 2 * cosh(4t) * sinh(4t) - 2 * sinh(4t) * cosh(4t)
Step 4: Simplify the expression.
In this case, we can observe that there is a common factor of 2 and also the terms cosh(4t) and sinh(4t) multiply to give sinh^2(t) - cosh^2(t) by the hyperbolic identity. Therefore, we can simplify the expression further:
f'(t) = 2 * (cosh(4t) * sinh(4t) - sinh(4t) * cosh(4t))
= 2 * sinh^2(4t) - 2 * cosh^2(4t)
So the simplified derivative of f(t) = cosh^2(4t) - sinh^2(4t) is f'(t) = 2 * sinh^2(4t) - 2 * cosh^2(4t).