If \(y = 2x^3 + 6x^2 + 6x + 1\), Find \(\frac{dy}{dx}\)
Question 1 Report
If \(y = 2x^3 + 6x^2 + 6x + 1\), Find \(\frac{dy}{dx}\)
Answer Details
The derivative of y with respect to x, denoted as dy/dx, is the rate at which y changes with respect to x. To find dy/dx, we need to apply the power rule and the sum rule for differentiation.
The power rule states that d(x^n)/dx = nx^(n-1). The sum rule states that d(f(x) + g(x))/dx = df(x)/dx + dg(x)/dx.
Given that y = 2x^3 + 6x^2 + 6x + 1, we can differentiate it term by term:
d(2x^3)/dx = 6x^2
d(6x^2)/dx = 12x
d(6x)/dx = 6
d(1)/dx = 0
Applying the sum rule, we have:
dy/dx = 6x^2 + 12x + 6
So, the answer is 6x^2 + 12x + 6.
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