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Chapter 3: Problem 50

Compute the derivative of the following functions. $$A=2500 e^{0.075 t}$$

Short Answer

Expert verified
Answer: The derivative of the given function with respect to \(t\) is \(\frac{dA}{dt} = 187.5 e^{0.075t}\).

Step by step solution

01

Identify the function to differentiate

The given function is \(A = 2500 e^{0.075 t}\), and we need to find its derivative with respect to \(t\).
02

Differentiate the function

To compute the derivative, we will apply the chain rule. First, differentiate the exponent of the exponential function, which is \(0.075t\). The derivative with respect to \(t\) is simply \(0.075\). Now, we will apply the formula mentioned in the analysis to differentiate \(e^{0.075t}\) with respect to its exponent. The derivative is \(0.075 e^{0.075t}\).
03

Apply the chain rule

The derivative of the outer exponential function was computed in Step 2. Then, the derivative of the entire function \((2500 e^{0.075t})\) with respect to \(t\) would be: $$\frac{dA}{dt} = 2500 (0.075 e^{0.075t})$$
04

Simplify the expression

To simplify the expression, multiply the constant terms (2500 and 0.075) together: $$\frac{dA}{dt} = 187.5 e^{0.075t}$$ #Phase 2: Final Answer# The derivative of the given function with respect to \(t\) is: $$\frac{dA}{dt} = 187.5 e^{0.075t}$$

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