Question

Find the first derivative and simplify the answer if possible. Use the notation from the lecture notes, NOT d/dx notation! State which rule or which combination of rules you are using other than power rule: Only mention product rule, quotient rule, chain rule! f(x)=−3e−⁴ˣ⁺²

Answer 1

To find the first derivative of the function f(x) = -3e^{-(4x+2)}, we apply the chain rule. Set u = -(4x+2) and differentiate the outer and inner functions. The derivative simplifies to f'(x) = 12e^{-(4x+2)}.

Step-by-step explanation:

The question asks us to find the first derivative of the function f(x) = -3e^{-(4x+2)} and to simplify the answer if possible. Using the chain rule, we differentiate the exponential function. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Let's set u = -(4x+2). Then f(x) = -3e^u. Applying the chain rule, we differentiate as follows:

1. Derivative of the outer function: e^u with respect to u is e^u.
2. Derivative of the inner function: u with respect to x is -4.
3. Multiply the derivative of the outer function by the derivative of the inner function and by the constant factor -3.

Putting it together, the derivative f'(x) becomes:

f'(x) = -3 * e^{u} * (-4) = 12e^{-(4x+2)}

This function has been simplified as it is a single term and cannot be simplified further.