YouLet y equals 8 x squared e to the power of 25 x end exponent.Find the differential, d y.A. d y equals 16 x e to the power of 25 x end exponent plus 200 x squared e to the power of 25 x end exponentB. d y equals left parenthesis 16 x e to the power of 25 x end exponent plus 200 x squared e to the power of 25 x end exponent right parenthesis d xC. d y equals 400 x e to the power of 25 x end exponent d xD. d y equals 16 x e to the power of 25 x end exponent d x
Question
YouLet y equals 8 x squared e to the power of 25 x end exponent.Find the differential, d y.A. d y equals 16 x e to the power of 25 x end exponent plus 200 x squared e to the power of 25 x end exponentB. d y equals left parenthesis 16 x e to the power of 25 x end exponent plus 200 x squared e to the power of 25 x end exponent right parenthesis d xC. d y equals 400 x e to the power of 25 x end exponent d xD. d y equals 16 x e to the power of 25 x end exponent d x
Solution
The function y is given as y = 8x^2 * e^(25x). To find the differential dy, we need to take the derivative of y with respect to x.
This involves using the product rule (d/dx [u*v] = u'v + uv') and the chain rule (d/dx[f(g(x))] = f'(g(x)) * g'(x)) of differentiation.
Let's set u = 8x^2 and v = e^(25x).
The derivative of u with respect to x (u') is 16x (using the power rule), and the derivative of v with respect to x (v') is 25e^(25x) (using the chain rule).
So, using the product rule, dy/dx = u'v + uv' = 16x * e^(25x) + 8x^2 * 25e^(25x) = 16x * e^(25x) + 200x^2 * e^(25x).
Therefore, the differential dy = (16x * e^(25x) + 200x^2 * e^(25x)) dx.
So, the correct answer is B. dy equals left parenthesis 16x * e to the power of 25x end exponent plus 200x^2 * e to the power of 25x end exponent right parenthesis dx.
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