Problem 7 - Calculus Single Variable

Question

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ f^{\prime}(c), f(x)=3 \sin (x)+4 \cos (x) $$

Step-by-Step Solution

Answer

The derivative is \( f'(c) = 3\cos(c) - 4\sin(c) \).

Step 1: Identify the Derivative Formula

To find the derivative of a trigonometric function, we use the derivatives of the basic trigonometric functions. The derivative of \( \sin(x) \) is \( \cos(x) \) and the derivative of \( \cos(x) \) is \(-\sin(x) \). The formula for finding the derivative of \( f(x) = 3\sin(x) + 4\cos(x) \) will involve these basic derivative rules.

Step 2: Apply the Constant Coefficient Rule

For any function \( af(x) \), where \( a \) is a constant, the derivative is \( a f'(x) \). Therefore, the derivative of \( 3\sin(x) \) is \( 3\cos(x) \), and the derivative of \( 4\cos(x) \) is \( 4(-\sin(x)) \).

Step 3: Calculate the Derivative

Using the results from the previous step, we combine the derivatives of each term: the derivative of \( 3\sin(x) \) is \( 3\cos(x) \) and of \( 4\cos(x) \) is \(-4\sin(x)\). Therefore, the derivative of the entire function is \( f'(x) = 3\cos(x) - 4\sin(x) \).

Step 4: Evaluate the Derivative at a Point

To find \( f'(c) \), where \( c \) is a specific point, simply substitute \( c \) for \( x \) in the derivative function. Thus, \( f'(c) = 3\cos(c) - 4\sin(c) \). This expression gives the slope of the tangent to the curve at \( x = c \).

Key Concepts

Trigonometric Derivatives

Understanding trigonometric derivatives is crucial when working with functions involving sine and cosine. The fundamental rules are simple yet powerful: the derivative of \( \sin(x) \) is \( \cos(x) \) and the derivative of \( \cos(x) \) is \(-\sin(x) \). These rules are derived from the limits and behaviors of these functions as angles change. Trigonometric derivatives help analyze how angles and arc lengths interact within the circle of the unit circle model, a key tool in understanding oscillations, waves, and signals in various fields of science and engineering.
When you're handling a function that is a combination of these trigonometric functions, remember that each part can be differentiated individually. This approach helps simplify complex expressions into smaller, manageable parts that can each follow these basic derivative rules.

Constant Coefficient Rule

The constant coefficient rule allows us to differentiate a function where a constant is multiplied by a trigonometric function efficiently. Essentially, this rule tells us that if we have a function of the form \( af(x) \), where \( a \) is a constant, its derivative is \( a \cdot f'(x) \).
This can be particularly useful because it breaks down the process by allowing us to "factor out" the constant and focus on the derivative of the actual function part. For example, in the expression \( 3\sin(x) \), the derivative becomes \( 3\cos(x) \), simply applying the derivative of \( \sin(x) \) and then multiplying by the constant 3. This rule simplifies the process of calculating derivatives when dealing with scaled functions.

Calculating Derivatives

When calculating derivatives, especially for functions involving a sum of terms like \( f(x) = 3\sin(x) + 4\cos(x) \), the process is systematic. You apply the derivative rules to each term separately.
Start by recognizing each part of the function: the function is composed of \( 3\sin(x) \) and \( 4\cos(x) \). Using known derivative rules:

• Calculate \( \frac{d}{dx}[3\sin(x)] \) which is \( 3\cos(x) \)
• Calculate \( \frac{d}{dx}[4\cos(x)] \) which is \( -4\sin(x) \)

Then combine these results to obtain the final derivative, \( f'(x) = 3\cos(x) - 4\sin(x) \). This systematic method is crucial for handling more complex functions where multiple rules might need to apply.

Tangent Slope Calculation

The derivative \( f'(x) \) does more than tell you how a function changes; it provides specific information about the tangent line at any given point on the function. The tangent slope calculation is straightforward: once you have the derivative, substitute the specific point \( x = c \) into the derivative.
For the function \( f(x) = 3\sin(x) + 4\cos(x) \), after finding \( f'(x) = 3\cos(x) - 4\sin(x) \), substituting \( c \) gives \( f'(c) = 3\cos(c) - 4\sin(c) \). This expression is the slope of the tangent line to the curve at \( x = c \).
Such calculations are invaluable in understanding behavior at specific points, useful in physics, engineering, and even economics, wherever rates of change at exact inputs need to be known.