Question

Find the first derivative for each in the form of dy/ dx, Show all work

y=3 e^{1}{2} x +2 x^{e^{3} 8) y=2 x^{3}- (2 x)^{5}

Answer 1

The derivatives of the given functions were calculated using the rules for differentiating exponentials and polynomials, resulting in dy/dx = (3e^{1/2}/2) e^{(1/2)x} + 2e^{38} x^{(e^{38}-1)} for the first function, and dy/dx = 6x^2 - 5(2x)^4 for the second function.

Step-by-step explanation:

The question asks for the first derivative of two functions, so we will apply the rules of differentiation to find dy/dx for each. For the first function y=3e^{1/2}x + 2x^{e^{38}}, we differentiate using the exponential rule and the power rule. For the second function y=2x^3 - (2x)^5, we again use the power rule, noting that in the second term we apply the chain rule to account for the composite function (2x).

For the first function, the derivative is dy/dx = (3e^{1/2}/2) e^{(1/2)x} + 2e^{38} x^{(e^{38}-1)}. For the second function, the derivative is dy/dx = 6x^2 - 5(2x)^4.