YouFind the derivative of y = 6x ^ 3 * e ^ (- 3x)A. d/dx (y) = - 54x ^ 2 * theta ^ (- 3x)B. d/dx (y) = 18x ^ 2 * theta ^ (- 3x)C. d/dx (y) = 18x ^ 2 * theta ^ (- 3x) - 18x ^ 3 * theta ^ (- 3x)D. d/dx (y) = - 18x ^ 2 * theta ^ (- 3x) + 18x ^ 3 * theta ^ (- 3x)
Question
YouFind the derivative of y = 6x ^ 3 * e ^ (- 3x)A. d/dx (y) = - 54x ^ 2 * theta ^ (- 3x)B. d/dx (y) = 18x ^ 2 * theta ^ (- 3x)C. d/dx (y) = 18x ^ 2 * theta ^ (- 3x) - 18x ^ 3 * theta ^ (- 3x)D. d/dx (y) = - 18x ^ 2 * theta ^ (- 3x) + 18x ^ 3 * theta ^ (- 3x)
Solution
To find the derivative of the function y = 6x^3 * e^(-3x), we will use the product rule and the chain rule. The product rule is (f*g)' = f'g + fg', and the chain rule is (f(g(x)))' = f'(g(x))*g'(x).
Let's denote f(x) = 6x^3 and g(x) = e^(-3x).
Step 1: Find f'(x) and g'(x) f'(x) = d/dx (6x^3) = 18x^2 g'(x) = d/dx (e^(-3x)) = -3e^(-3x) (using the chain rule)
Step 2: Apply the product rule d/dx (y) = f'(x)*g(x) + f(x)*g'(x) = 18x^2 * e^(-3x) - 18x^3 * e^(-3x)
So, the correct answer is C. d/dx (y) = 18x^2 * e^(-3x) - 18x^3 * e^(-3x).
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