Question

Use the General Power Rule where appropriate to find the derivative of the following functions. $$s(t)=\cos 2^{t}$$

Short answer

Question: Find the derivative of the function s(t) = cos(2^t) with respect to t. Answer: The derivative of the function s(t) = cos(2^t) with respect to t is s'(t) = -sin(2^t) * (2^t * ln(2)).

Step-by-step solution

Identify the inner function

The inner function is the function inside the cosine, which is 2^t.

Find the derivative of the inner function

The inner function is g(t) = 2^t. We can find its derivative g'(t) using the exponential function rule (d(e^u)/dt = e^u*(du/dt)): $$g'(t) = (2^t) \cdot \ln{2}$$

Apply the chain rule

The chain rule states that if we have a composite function, the derivative of the composite function is equal to the derivative of the outer function times the derivative of the inner function. In this case, our outer function is f(u) = cos(u) and the inner function is g(t) = 2^t, so s(t) = f(g(t)). Let's find the derivative of the outer function: $$f'(u) = -\sin(u)$$ Now we can use the chain rule to find s'(t): $$s'(t) = f'(g(t)) \cdot g'(t) = -\sin(2^t) \cdot (2^t \cdot \ln{2})$$

Write the final answer

The derivative of the function s(t) = cos(2^t) with respect to t is: $$s'(t) = -\sin(2^t) \cdot (2^t \cdot \ln{2})$$

Key concepts

General Power Rule

The General Power Rule is a fundamental concept in calculus used to differentiate functions of the form \((u(x))^n\). It extends the basic power rule, which is only applicable to simple power functions like \(x^n\). Here, \(u(x)\) represents any differentiable function, and \(n\) is a real number that dictates the power to which \(u(x)\) is raised.

When using the General Power Rule, follow these steps:

• Differentiate the outer function as usual, treating the inner function \(u(x)\) as a constant. This means applying the standard power rule: bring down the exponent and reduce it by one.
• Multiply by the derivative of the inner function \(u'(x)\).

Putting it into a formula, if you have a function \(y = (u(x))^n\), then the derivative is given by the rule:
\[ \frac{dy}{dx} = n \cdot (u(x))^{n-1} \cdot u'(x)\]

In our example, the General Power Rule wasn't directly used, but it is fundamental in understanding how to handle expressions where an inner function might be raised to a power.

Chain Rule

The Chain Rule is one of the cornerstones of differential calculus. It is particularly useful when dealing with composite functions, where you have a function nested inside another function, often making them seemingly complex to differentiate at first glance.

The principle behind the Chain Rule is simple: differentiate the outer function first, leaving the inner function as it is, and then multiply it by the derivative of the inner function. The Chain Rule is expressed in mathematical terms as:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]
Here, \(f\) is the outer function, \(g(x)\) is the inner function, and \(g'(x)\) is the derivative of the inner function.

In the context of our example, we used the Chain Rule to differentiate \(s(t) = \cos(2^t)\). The outer function is \(f(u) = \cos(u)\) and the inner function is \(g(t) = 2^t\). After identifying these, we take the derivative of the outer function, which is \(-\sin(u)\), and multiply it by the derivative of the inner function, \((2^t \ln(2))\).

This step-by-step approach solidifies how the Chain Rule effectively breaks down complex derivatives into more manageable pieces.

Exponential Functions

Exponential functions are an intriguing type of function where a constant base is raised to a variable exponent, usually taking the form \(a^x\), where \(a\) is a positive constant and \(x\) is the variable. These functions are essential in calculus due to their constant growth rate properties, which have significant applications in fields such as finance, biology, and physics.

The derivative of an exponential function is closely tied to the natural logarithm. For a general exponential function \(g(x) = a^x\), the derivative is:
\[ g'(x) = a^x \cdot \ln(a)\]
This formula reflects the rate of change of an exponential function being proportional to its current value—a unique property of exponentials.

In the example problem, identifying and differentiating the inner function \(2^t\) was a crucial step. Here, using the rule for exponential functions, we found that the derivative was \((2^t \cdot \ln(2))\), showing how exponential functions smoothly integrate with other calculus rules like the Chain Rule for determining derivatives of composite functions.