Find the nth derivative of sinh^2x . Cosh^ 2x
To find the nth derivative of the function f(x) = sinh^2x * cosh^2x, we can use the chain rule and the product rule of differentiation.
Let's start by finding the first derivative.
f'(x) = [(sinh^2x)' * cosh^2x] + [sinh^2x * (cosh^2x)']
Using the chain rule, we can differentiate sinh^2x and cosh^2x separately.
(sin^2hx)' = 2sinhx * (sinhx)'
(cosh^2x)' = 2coshx * (coshx)'
Now we can substitute these values back into the first derivative equation.
f'(x) = [2sinhx * (sinhx)' * cosh^2x] + [sinh^2x * 2coshx * (coshx)']
f'(x) = 2sinhx * coshx * cosh^2x + sinh^2x * 2coshx * sinh^2x
To find the second derivative, we differentiate f'(x) using the product rule again.
f''(x) = [2sinhx * coshx * cosh^2x]' + [sinh^2x * 2coshx * sinh^2x]'
Differentiating each term separately:
(2sinhx * coshx * cosh^2x)' = (2sinhx)' * coshx * cosh^2x + 2sinhx * (coshx)' * cosh^2x + 2sinhx * coshx * (cosh^2x)'
(sinh^2x * 2coshx * sinh^2x)' = (sinh^2x)' * 2coshx * sinh^2x + sinh^2x * (2coshx)' * sinh^2x + sinh^2x * 2coshx * (sinh^2x)'
Again, using the chain rule:
(2sinhx)' = 2coshx
(coshx)' = sinh^2x
(sinh^2x)' = 2sinhx
(cosh^2x)' = 2sinhx * coshx
Now we can substitute these values back into the second derivative equation.
f''(x) = 2coshx * coshx * cosh^2x + 2sinhx * sinh^2x * cosh^2x + 2sinhx * coshx * sinh^2x + 2sinhx * coshx * 2sinhx * coshx
You can continue this process to find higher order derivatives if needed. Each time, you will need to apply the product rule and the chain rule to differentiate the terms.