Problem 43 - Single Variable Calculus

Question

Find the derivative of the function. $$ y=e^{\cos x} $$

Step-by-Step Solution

Answer

The derivative of the function \(y = e^{\cos x}\) is \(\frac{dy}{dx} = -\sin x \cdot e^{\cos x}\).

Step 1: Identify the outer and inner functions

Outer function: \(y = e^u\) (exponential function) Inner function: \(u = \cos x\) (cosine function)

Step 2: Find the derivatives of the outer and inner functions

Derivative of the outer function: \(\frac{dy}{du} = e^u\) Derivative of the inner function: \(\frac{du}{dx} = -\sin x\)

Step 3: Apply the chain rule

According to the chain rule, the derivative of the composite function is the product of the derivatives of the outer and inner functions: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

Step 4: Substitute the derivatives from Step 2

\(\frac{dy}{dx} = e^u \cdot (-\sin x)\)

Step 5: Substitute the inner function back into the equation

Since \(u = \cos x\), we substitute it back into the equation to obtain the final derivative: \(\frac{dy}{dx} = e^{\cos x} \cdot (-\sin x)\)

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. When you have a function nested within another, like a chain, the chain rule helps unravel the layers. For instance, in our given problem, the function is composed of an exponential function with a trigonometric function inside it, i.e., \( y = e^{\cos x} \).

Here's how the chain rule operates:

• Identify the outer function, which is the "container", in this case, \( y = e^u \), where \( u = \cos x \).
• Then recognize the inner function, which is "inside" the container, \( u = \cos x \).
• Next, take the derivative of both functions separately: the outer with respect to its inside and the inner with respect to \( x \).
• Finally, multiply these derivatives to get the derivative of the composite function.

By applying the chain rule, we calculate that the derivative of \( y = e^{\cos x} \) is given by \( e^{\cos x} \cdot (-\sin x) \). A simple yet powerful technique that efficiently breaks down complex derivatives into manageable pieces.

Exponential Function

An exponential function is one where the variable is in the exponent, meaning it takes the form \( a^x \) or in our specific case, \( e^x \), where \( e \) is the natural exponential base (approximately 2.718). These functions are known for their rapid growth or decay behavior. In our exercise, \( y = e^{\cos x} \), the exponential function is vital because of the natural exponential function's unique derivative property.

Key attributes of exponential functions:

• The derivative of \( e^x \) is itself, \( e^x \), which makes it distinct and straightforward in calculus.
• Exponential functions are continuous and differentiable everywhere.
• They frequently appear in problems involving growth, decay, and natural processes.

The understanding of exponential functions in calculus is crucial as it simplifies the differentiation process, especially when combined with other functions via the chain rule as seen.

Trigonometric Function

Trigonometric functions describe relationships between the angles and sides of triangles. Commonly used trigonometric functions include sine, cosine, and tangent. Here, our focus is on the cosine function, \( \cos x \), which forms the inner part of the composite function \( y = e^{\cos x} \).

Important aspects of trigonometric functions:

• Cosine has a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.
• The derivative of the cosine function is \( -\sin x \), a key aspect used in applying the chain rule.
• Trigonometric identities often simplify complex expressions, facilitating easier differentiation.

Understanding how trigonometric functions like cosine interact with other mathematical functions is essential for calculus students, particularly when differentiating composite functions as in our exercise.