Question

Find the derivative of the following functions. $$A=250(1.045)^{4 t}$$

Short answer

Question: Find the derivative with respect to \(t\) of the function \(A(t) = 250(1.045)^{4t}\). Answer: The derivative of the function with respect to \(t\) is \(\frac{dA}{dt} = 1000\ln(1.045)(1.045)^{4t}\).

Step-by-step solution

1. Identify the Inner and Outer Functions

Identify the inner function and the outer function for the chain rule. In this case, the inner function is \(u(t) = 4t\), and the outer function is \(A(u) = 250(1.045)^u\).

2. Derivative of the Inner Function

Determine the derivative of the inner function with respect to \(t\). In this case, \(\frac{du}{dt} = \frac{d(4t)}{dt} = 4\).

3. Derivative of the Outer Function

Determine the derivative of the outer function with respect to the inner function (\(u\)). In this case, \(\frac{dA}{du} = \frac{d(250(1.045)^u)}{du} = 250\ln(1.045)\times(1.045)^u\).

4. Apply the Chain Rule

Apply the chain rule by multiplying the derivatives of the inner and outer functions. So, the derivative of the given function with respect to \(t\) is \(\frac{dA}{dt} = \frac{dA}{du}\times\frac{du}{dt} = (250\ln(1.045)\times(1.045)^u)\times 4\).

5. Substitute the Inner Function

Substitute the inner function \(u(t) = 4t\) back into the expression for the derivative. Therefore, the derivative of the given function with respect to \(t\) is \(\frac{dA}{dt} = 250 \times 4\ln(1.045)(1.045)^{4t} = 1000\ln(1.045)(1.045)^{4t}\).

Key concepts

Derivative

In calculus, a derivative represents how a function changes as its input changes. It essentially measures the rate at which something is changing at a particular instant. This is crucial in understanding behaviors in real-world scenarios. For example, in our exercise, we are interested in finding out how the function \(A=250(1.045)^{4t}\) changes as the variable \(t\) changes. This helps in assessing the growth rate of the function over time. To determine the derivative of a function like this one, we often need to use tools like the chain rule, particularly when the function is composed of other functions nested within each other. Breaking down complex expressions into simpler parts allows mathematicians to analyze and comprehend them more easily.

Inner Function

An inner function is part of a composite function or any function that lives inside another function. In our exercise, we identified the inner function as \(u(t) = 4t\). Inner functions are fundamental when applying the chain rule because the derivative of the overall function depends on this part.

• Think of the inner function as the first layer or building block on which the entire expression depends.
• To find its derivative, simply observe its relationship with the variable of interest, here being \(t\).

In our example, computing the derivative of the inner function with respect to \(t\) was straightforward: \(\frac{du}{dt} = 4\). This step lays the groundwork for using the chain rule effectively.

Outer Function

The outer function is the function that wraps around the inner function. In our current exercise, it is \(A(u) = 250(1.045)^u\). The outer function relies on the inner function's output to determine its own output. Hence, understanding its behavior is key when unraveling composite functions.

• It's like an external layer that might add complexity to the simple result given by the inner function.
• Finding its derivative with respect to the inner function, denoted by \(u\) in this case, involves considering it as a function independently of the original variable \(t\).

For our function, this derivative was \(\frac{dA}{du} = 250\ln(1.045)\times(1.045)^u\). Mastery in differentiating such functions comes through practice.

Exponential Function

Exponential functions are crucial in mathematics due to their consistent growth rates. They are of the form \(a\times b^x\), where \(a\) is a constant and \(b\) is the base of the exponential. In our problem, the exponential function is represented by \(250(1.045)^{4t}\).

• Exponential functions can model real-world phenomena like population growth or radioactive decay.
• The growth factor \(1.045\), known as the base, indicates the constant rate of growth per time unit \(t\).

During differentiation, the exponential behavior is retained, recognizable by the pattern of the derivative, which includes the natural logarithm of the base. This key characteristic helps in predicting how quantities grow or shrink over time. Therefore, understanding and differentiating exponential functions is a vital skill in calculus and its applications.