find the derivative of y with respect to x if y=(13x^2-26x+26)e^-3x. y'=
To find the derivative of y with respect to x, you can use the product rule and the chain rule. Let's break down the process step by step:
Step 1: Apply the product rule
The product rule states that if you have two functions u(x) and v(x), then the derivative of their product, y = u(x) * v(x), is given by the formula:
y' = u'v + uv'
In this case, we can consider u(x) = (13x^2 - 26x + 26) and v(x) = e^(-3x).
Step 2: Calculate the derivatives of u(x) and v(x)
The derivative of u(x) = (13x^2 - 26x + 26) can be found using the power rule and the constant rule:
u'(x) = d/dx (13x^2) - d/dx (26x) + d/dx (26) = 26x - 26
The derivative of v(x) = e^(-3x) can be found using the chain rule:
v'(x) = d/dx (e^(-3x)) = e^(-3x) * (-3) = -3e^(-3x)
Step 3: Substitute the calculated derivatives into the product rule formula
y' = u'v + uv'
= (26x - 26) * e^(-3x) + (13x^2 - 26x + 26) * (-3e^(-3x))
= 26x * e^(-3x) - 26 * e^(-3x) - 39x^2 * e^(-3x) + 78x * e^(-3x) - 78 * e^(-3x)
Therefore, the derivative of y with respect to x, y', is given by:
y' = 26x * e^(-3x) - 26 * e^(-3x) - 39x^2 * e^(-3x) + 78x * e^(-3x) - 78 * e^(-3x)
y' = (26x^2-26)e^-3x - 3(13x^2-26x+26)e^-3x
= -13(3x^2-8x+8)e^-3x