find f'(x) if f(x)=x^2e^(x(3x-1)^3
To find the derivative of the given function f(x) = x^2e^(x(3x-1)^3), we can use the chain rule and product rule of differentiation.
Let's break down the function into its individual components:
f(x) = g(x) * h(x)
where g(x) = x^2 and h(x) = e^(x(3x-1)^3).
Now, let's find the derivatives of g(x) and h(x) separately:
Derivative of g(x):
g'(x) = 2x (using the power rule of differentiation)
Derivative of h(x):
To find the derivative of e^(x(3x-1)^3), we need to use the chain rule. Let's break it down further:
Let u = (3x-1)^3
h(x) = e^(xu), where u = (3x-1)^3
Now, let's find the derivative of h(x) with respect to x using the chain rule:
h'(x) = du/dx * e^u
To find du/dx, we differentiate u with respect to x:
du/dx = 3(3x-1)^2 * d(3x-1)/dx
d(3x-1)/dx = 3 (since the derivative of 3x with respect to x is 3 and the derivative of -1 with respect to x is 0)
Therefore, du/dx = 3(3x-1)^2 * 3
Substituting this back into the h'(x) equation:
h'(x) = 9(3x-1)^2 * e^u
Now, we have the derivatives of g(x) and h(x). To find the derivative of f(x), we can use the product rule:
f'(x) = g'(x) * h(x) + g(x) * h'(x)
Substituting the derivatives we found earlier:
f'(x) = (2x) * e^(x(3x-1)^3) + (x^2) * 9(3x-1)^2 * e^u
Finally, substituting back u = (3x-1)^3:
f'(x) = 2xe^(x(3x-1)^3) + 9x^2(3x-1)^2 * e^(x(3x-1)^3)