Problem 4 - Applied Calculus

Question

Differentiate each function. $$ f(t)=(1+\sin t)^{4} $$

Step-by-Step Solution

Answer

The derivative is \(f'(t) = 4\cos t (1+\sin t)^3\).

Step 1: Identify the Function and the Rule

The function given is \(f(t) = (1 + \sin t)^4\). This is a composite function because it involves a power and a trigonometric function inside. We will use the chain rule to differentiate it.

Step 2: Apply the Chain Rule

Using the chain rule, we know that \(\frac{d}{dt} [u(t)^n] = n\cdot u(t)^{n-1}\cdot u'(t)\) where \(u(t) = 1 + \sin t\) and \(n=4\). First, differentiate the outer function: \(4(1+\sin t)^3\).

Step 3: Differentiate the Inner Function

The inner function is \(u(t) = 1 + \sin t\). Differentiate it with respect to \(t\) to get \(u'(t) = \cos t\) since the derivative of \(\sin t\) is \(\cos t\).

Step 4: Combine Results Using the Chain Rule

Now combine the results using the chain rule: \[ f'(t) = 4(1+\sin t)^3 \cdot \cos t \]This is the derivative of the function \(f(t)\).

Step 5: Simplify if Needed

While simplification is not strictly necessary, ensure the expression is clear. Thus, the derivative is readily expressed as:\[ f'(t) = 4\cos t (1+\sin t)^3 \]

Key Concepts

Composite Function

A composite function is essentially a function within another function. This concept is crucial when you have an equation where one function's value is plugged into another. For instance, in the exercise where we are given the function \( f(t) = (1 + \sin t)^4 \), there is an inner function \( u(t) = 1 + \sin t \) and an outer function \( (u)^4 \).

We can think of the composite function as a two-step process:

• First, calculate the inner function \( u(t) = 1 + \sin t \) for a given \( t \).
• Then, take the value of the inner function and apply the outer operation, \( (u)^4 \). This means you raise it to the power of 4.

This composition of functions is what makes the process of differentiation slightly more involved and requires specific techniques like the chain rule to solve effectively.

Chain Rule

The chain rule is a fundamental differentiation rule used for finding the derivative of composite functions. It provides a way to handle the complexity introduced by the composition of two or more functions.

Here's how the chain rule works in simpler terms:- Imagine you want to differentiate a function \( g(f(x)) \).- The chain rule states that the derivative of this function is \( g'(f(x)) \cdot f'(x) \).When applying this to our function \( f(t) = (1 + \sin t)^4 \),

• First, differentiate the outer function \( g(u) = u^4 \), giving us \( 4(1+\sin t)^3 \).
• Next, differentiate the inner function \( u(t) = 1 + \sin t \), which results in \( \cos t \).
• Finally, multiply these two derivative results together: \( f'(t) = 4(1+\sin t)^3 \cdot \cos t \).

The chain rule simplifies handling these nested functions and is indispensable when differentiating composite functions.

Trigonometric Function Differentiation

Differentiating trigonometric functions is an essential skill in calculus, particularly because these functions often appear as components in more complex functions. The trigonometric functions — sine, cosine, tangent, etc. — have well-known derivatives:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

In the given function \( f(t) = (1 + \sin t)^4 \), the inner function incorporates the sine function. Applying the rule for differentiating sine, the derivative of \( \sin t \) becomes \( \cos t \).

Understanding how to differentiate these trigonometric functions allows us to apply them effectively within the larger context of composite functions and the chain rule, ensuring we can solve a variety of calculus problems accurately.