Question

Compute the derivative of the given function. $$f(\theta)=9 \sin \theta+10 \cos \theta$$

Short answer

The derivative is \( f'(\theta) = 9 \cos \theta - 10 \sin \theta \).

Step-by-step solution

Identify the function's components

The function given is \( f(\theta) = 9 \sin \theta + 10 \cos \theta \). Notice that it is composed of two basic trigonometric functions, \( \sin \theta \) and \( \cos \theta \), each multiplied by constants 9 and 10 respectively.

Recall derivative rules for basic trig functions

The derivative of \( \sin \theta \) with respect to \( \theta \) is \( \cos \theta \) and the derivative of \( \cos \theta \) with respect to \( \theta \) is \(-\sin \theta \). We will use these rules to differentiate each component of the function.

Differentiate each component separately

Differentiate \( 9 \sin \theta \) to get \( 9 \cos \theta \). Differentiate \( 10 \cos \theta \) to get \( -10 \sin \theta \). Use the constant multiplication rule which states that the derivative of a constant times a function is the constant times the derivative of the function.

Combine the derivatives

Add the derivatives found in Step 3 together: \( 9 \cos \theta + (-10 \sin \theta) \). Simplify it to \( 9 \cos \theta - 10 \sin \theta \).

Write the final derivative

The derivative of the function \( f(\theta) = 9 \sin \theta + 10 \cos \theta \) with respect to \( \theta \) is \( f'(\theta) = 9 \cos \theta - 10 \sin \theta \).

Key concepts

Derivative

The concept of a derivative is foundational in calculus. It represents the rate at which a function is changing at any given point. To imagine it visually, consider a line tangent to a curve at a particular point. The slope of this tangent line is the derivative at that point. Derivatives help us understand how one quantity changes in relation to another and provide critical insights into the behavior of functions. They are widely used in various fields such as physics, economics, and biology to model dynamic processes.

• Instantaneous Rate of Change: Derivatives represent how a function changes instantaneously with respect to its variable. For instance, in physics, it could indicate the speed, showing how position changes over time.
• Finding Tangent Lines: The derivative at a given point is the slope of the tangent to the function at that point, providing a linear approximation of the function near the point.
• Notations: Derivatives are denoted by different symbols such as \( f'(x) \), \( \frac{dy}{dx} \), or \( \frac{df}{dx} \), all representing the same concept of differentiation.

Trigonometric Functions

Trigonometric functions such as sine and cosine are essential in many areas of mathematics and applied sciences. These functions describe relationships in right-angled triangles and oscillatory motions like waves. For the function \( f(\theta) = 9 \sin \theta + 10 \cos \theta \), we see how trigonometric functions can be combined to form more complex expressions. Trigonometric functions are periodic, and their derivatives maintain this cyclical behavior.

• Definitions: Sine and cosine functions can be defined using the unit circle. \( \sin \theta \) represents the y-coordinate and \( \cos \theta \) represents the x-coordinate of a point on the circle.
• Periodicity: Both sine and cosine functions repeat their values in a regular pattern. This property is what makes them useful for modeling periodic phenomena such as sound waves or rotations.
• Applications: Beyond geometry, trigonometric functions are crucial in signal processing, electrical engineering, and even in predicting stock market trends.

Differentiation Rules

Differentiation rules are guidelines that simplify the process of finding derivatives, especially for complex functions. When dealing with trigonometric functions like \( \sin \theta \) and \( \cos \theta \), specific rules apply. These crucial rules allow us to break down functions into simpler parts for easier differentiation.

• Basic Trigonometric Derivatives: The derivative of \( \sin \theta \) is \( \cos \theta \), while the derivative of \( \cos \theta \) is \(-\sin \theta \). Knowing these basics allows us to quickly find derivatives of functions involving sine and cosine.
• Constant Multiplication Rule: If a function is multiplied by a constant, its derivative is simply the derivative of the function multiplied by that constant. For \( 9 \sin \theta \), we multiply \( 9 \) by the derivative of \( \sin \theta \), which is \( \cos \theta \).
• Sum Rule: When a function is the sum of several terms, the derivative of the function is the sum of the derivatives of each term. Thus, for \( 9 \cos \theta - 10 \sin \theta \), you find the derivative of each part separately and add them together.