Find the derivative of (x^2)(e^-3x)
To find the derivative of the given function, which is a product of two functions, we can use the product rule. The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In this case, u(x) = x^2 and v(x) = e^(-3x).
Now let's find the derivatives of u(x) and v(x):
The derivative of u(x) = x^2 with respect to x is obtained by applying the power rule, which states that if we have a function u(x) = x^n, then the derivative is given by:
u'(x) = n * x^(n-1)
So, in this case, the derivative of u(x) = x^2 is:
u'(x) = 2 * x^(2-1) = 2x
Next, let's find the derivative of v(x) = e^(-3x).
The derivative of v(x) = e^(-3x) can be found using the chain rule, which states that if we have a function v(x) = f(g(x)), then the derivative is given by:
v'(x) = f'(g(x)) * g'(x)
In this case, f(x) = e^x and g(x) = -3x.
The derivative of f(x) = e^x is simply f'(x) = e^x.
The derivative of g(x) = -3x is g'(x) = -3.
So, substituting these derivatives into the chain rule formula, we get:
v'(x) = e^(-3x) * (-3)
Now, applying the product rule with u'(x) = 2x and v'(x) = e^(-3x) * (-3), we have:
(d/dx)((x^2)(e^-3x)) = (2x)(e^-3x) + (x^2)(e^(-3x) * (-3))
Simplifying this expression gives us the final result, which is the derivative of the given function.