Derivative of:
(exp(8x^4))(5x^3+9)^3
___________________________ (divided by)
(22x^2+4x-20)^2
To find the derivative of the given expression, we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x)/g(x), where f(x) and g(x) are both differentiable functions, then the derivative is given by:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
In our case, f(x) = (exp(8x^4))(5x^3+9)^3 and g(x) = (22x^2+4x-20)^2.
To find the derivative, we need to find the derivatives of f(x) and g(x) separately and then apply the quotient rule.
Let's start with finding the derivative of f(x):
Using the chain rule, we have:
f'(x) = (exp(8x^4))(5x^3+9)^3 * d/dx(8x^4)
= (exp(8x^4))(5x^3+9)^3 * (32x^3)
Next, let's find the derivative of g(x):
Using the power rule, we have:
g'(x) = 2(22x^2+4x-20) * d/dx(22x^2+4x-20)
= 2(22x^2+4x-20) * (44x+4)
Now that we have the derivatives of f(x) and g(x), we can apply the quotient rule:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
= ((exp(8x^4))(5x^3+9)^3 * (32x^3) * (22x^2+4x-20)^2 - (exp(8x^4))(5x^3+9)^3 * (22x^2+4x-20) * (44x+4)) / ((22x^2+4x-20)^2)^2
Simplifying this expression further may require additional algebraic steps depending on the specific values of x.
Note: The complexity of this derivative may make the calculation cumbersome. It may be helpful to use computational software or online calculators to obtain the final result.