Final answer:
To find the derivative of y = (e^{3x})(x^3) + 2x - 5, we apply the product rule and power rule. The result is y' = 3e^{3x}x^3 + e^{3x}3x^2 + 2.
Step-by-step explanation:
Finding the Derivative of a Function
To find the derivative of the function y = (e^{3x})(x^3) + 2x - 5, we use the product rule and power rule of differentiation. According to the product rule, if we have a function that is the product of two functions, say u(x) and v(x), the derivative is u'(x)v(x) + u(x)v'(x). We also use the power rule which states that the derivative of x^n is n*x^(n-1).
Using these rules, we derive step by step:
First, let's set u(x) = e^{3x} and v(x) = x^3. We find the derivatives u'(x) = 3e^{3x} (since the derivative of e^{kx} is ke^{kx}) and v'(x) = 3x^2 (applying the power rule).
Now we apply the product rule:
y' = u'(x)v(x) + u(x)v'(x) = 3e^{3x}x^3 + e^{3x}3x^2. We also differentiate the rest of the function which is a linear function and its derivative is simply its coefficient. Therefore, the derivative of 2x - 5 is 2.
Putting it all together we get:
y' = 3e^{3x}x^3 + e^{3x}3x^2 + 2