Find The Derivative
y = 5x^(2)e^(3x)
Please Show Work
Just use the product rule and the chain rule:
y = 5x2e3x
y' = 5 * 2x e3x + 5x2 * e3x * 3
y' = 5xe3x(2 + 3x)
Good
I don't understand how you got the answer. Can you break it down further with more detail?
To find the derivative of the function y = 5x^2e^(3x), we can use the product rule and the chain rule of differentiation.
Using the product rule, the derivative of y with respect to x is given by:
dy/dx = (d/dx)(5x^2)(e^(3x)) + (d/dx)(e^(3x))(5x^2)
To differentiate the first term, we apply the power rule, which states that the derivative of x^n with respect to x is given by nx^(n-1). Therefore, the derivative of 5x^2 with respect to x is 10x^1, which simplifies to 10x.
Differentiating the second term requires the chain rule, which states that if we have a composition of functions, we need to multiply by the derivative of the inner function. Here, the inner function is 3x, and the derivative of e^(3x) with respect to x is e^(3x)(3), which simplifies to 3e^(3x).
The derivative of y with respect to x is:
dy/dx = 10x * e^(3x) + 3e^(3x) * 5x^2
Simplifying this expression further, we have:
dy/dx = 10xe^(3x) + 15x^2e^(3x)
So the derivative of y = 5x^2e^(3x) is dy/dx = 10xe^(3x) + 15x^2e^(3x).