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 for a function of the form f(x)/g(x), the derivative is given by:
f'(x)g(x) - f(x)g'(x)
_________________
[g(x)]^2
Let's break down the given expression using the quotient rule:
f(x) = (exp(8x^4))(5x^3+9)^3
g(x) = (22x^2+4x-20)^2
Now, let's find the derivatives of f(x) and g(x).
To find f'(x), we need to use the chain rule since f(x) is composed of two functions: exp(8x^4) and (5x^3+9)^3.
Let's start with the outer function first: (5x^3+9)^3
Using the chain rule, we get:
[3(5x^3+9)^2] * (15x^2)
Next, let's find the derivative of the inner function, exp(8x^4):
Using the chain rule, we get:
(exp(8x^4)) * 32x^3
Now, we can find f'(x):
f'(x) = (exp(8x^4))(32x^3) + (exp(8x^4)) * [3(5x^3+9)^2] * (15x^2)
Moving on to g'(x), we simply need to find the derivative of (22x^2+4x-20)^2:
Using the chain rule, we get:
[2(22x^2+4x-20)] * (44x+4)
Now, we have f'(x) and g'(x). We can substitute them back into the quotient rule formula:
[f'(x)g(x) - f(x)g'(x)]
_________________
[g(x)]^2
Substituting in the values, we get:
[(exp(8x^4))(32x^3)(22x^2+4x-20)^2 + (exp(8x^4)) * [3(5x^3+9)^2] * (15x^2)(22x^2+4x-20)]
__________________________________________________________________________________________
[g(x)]^2
Simplifying this expression gives us the derivative of the given expression.