Question

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d x} \cos F(x) $$

Short answer

The derivative is \(-\sin(F(x)) \cdot F'(x)\).

Step-by-step solution

Identify the Problem

We need to find the derivative of the expression \( \cos F(x) \) with respect to \( x \). Since \( F(x) \) is a differentiable function, we will use the chain rule.

Apply the Chain Rule

The chain rule is used when differentiating compositions of functions. In this exercise, let \( u = F(x) \). The derivative of \( \cos u \) with respect to \( x \) using the chain rule is \(-\sin u \cdot \frac{du}{dx}\).

Differentiate \( \cos F(x) \)

First, differentiate \( \cos u \) with respect to \( u \). The derivative is \(-\sin u\). Then, multiply by the derivative of \( u = F(x) \) with respect to \( x \), which is \( F'(x) \). This gives the expression \(-\sin(F(x)) \cdot F'(x)\).

Combine and Simplify

Putting it all together, the expression for the derivative is \(-\sin(F(x)) \cdot F'(x)\). This expresses the derivative in terms of the function \( F(x) \) and its derivative.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus, used to understand the rate at which values of a function change. When we differentiate a function, we're essentially looking to find its derivative. This involves determining how a small change in the input of a function will affect its output.

To differentiate is to analyze how functions behave when their variables change. For example, if you have a function that gives the position of a car over time, its derivative will tell you how fast the car is going at any given moment – essentially, the car's speed. Differentiation has many practical applications, such as finding the slope of a curve, the velocity of an object, and optimizing functions.

In the exercise above, differentiation is applied to the composition of functions using the chain rule. This method allows us to find the derivative of a composite function by evaluating the derivative of the outer function and multiplying it by the derivative of the inner function. All these elements show the use of differentiation to ease understanding complex mathematical functions.

Function Composition

Function composition is the process of applying one function to the results of another. It's like running the outputs of one function through another function. If you think of a function as a machine with an input and an output, composing functions is like feeding the output of one machine directly into the input of another.

In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), the composition \( f(g(x)) \) means you're applying \( f \) to \( g(x) \). To solve differentiable problems involving composition like in our exercise, it's important to recognize how these functions are nested within each other. In the case of \( \cos F(x) \), the function \( F(x) \) is nested inside the cosine function. This situation calls for the chain rule, a technique that simplifies differentiating complex compositions.

Understanding how to unravel these compositions and how to manage them with rules like the chain rule is crucial in calculus. It allows us to tackle a wide range of real-world problems by breaking down complex processes into simpler, manageable parts.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are used widely in mathematics to model periodic phenomena. These functions relate the angles of triangles to the lengths of their sides in the context of the unit circle. They're vital in many fields, including physics, engineering, and computer science.

In differentiation, these functions have specific rules to follow. For example, the derivative of \( \cos(x) \) is \(-\sin(x)\). This forms part of the exercise solution where \( \cos(F(x)) \) requires differentiation. It highlights the use of trigonometric identities in deriving new functions.

These functions are particularly useful because they can represent waves and oscillations – anything that happens over cycles or repeats at regular intervals can often be modeled using trigonometry. Learning and applying the derivatives of trigonometric functions is, therefore, a fundamental aspect of understanding more advanced calculus concepts.