compute the derivative of the following function. f(x)=5xe^{6x} f(x)=□

Explanation:

Step1: Apply product - rule

The product - rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Here, let (u = 5x) and (v = e^{6x}).

Step2: Find the derivative of (u)

The derivative of (u = 5x) with respect to (x) is (u^\prime=\frac{d}{dx}(5x)=5).

Step3: Find the derivative of (v)

Using the chain - rule, if (v = e^{6x}), let (t = 6x), then (\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}). Since (\frac{d}{dt}(e^{t}) = e^{t}) and (\frac{dt}{dx}=6), we have (v^\prime=\frac{d}{dx}(e^{6x})=6e^{6x}).

Step4: Calculate (f^\prime(x))

By the product - rule (f^\prime(x)=u^\prime v+uv^\prime). Substitute (u = 5x), (u^\prime = 5), (v = e^{6x}), and (v^\prime = 6e^{6x}) into the formula: (f^\prime(x)=5\cdot e^{6x}+5x\cdot6e^{6x}=5e^{6x}(1 + 6x)).

Answer:

(5e^{6x}(1 + 6x))