Problem 17 - Calculus

Question

Find the derivatives of the functions $$ e^{\sin x}+\sin e^{x} $$

Step-by-Step Solution

Answer

The derivative is \( e^{\sin x} \cos x + \cos e^{x} \cdot e^x \).

Step 1: Identify Each Function

The given function is composed of two parts: \( e^{\sin x} \) and \( \sin e^{x} \). You need to find the derivative of each part separately and then add them together.

Step 2: Differentiate \( e^{\sin x} \) Using the Chain Rule

The derivative of \( e^{\sin x} \) with respect to \( x \) requires the chain rule. First, differentiate the outer function \( e^u \) with respect to \( u \) (where \( u = \sin x \)), which gives \( e^{\sin x} \). Then differentiate \( \sin x \) with respect to \( x \), giving \( \cos x \). Thus, the derivative of \( e^{\sin x} \) is \( e^{\sin x} \cos x \).

Step 3: Differentiate \( \sin e^{x} \) Using the Chain Rule

For \( \sin e^{x} \), use the chain rule again. First, differentiate the outer \( \sin u \) with respect to \( u \) (where \( u = e^x \)), giving \( \cos e^{x} \). Then differentiate \( e^x \) with respect to \( x \), giving \( e^x \). Thus, the derivative of \( \sin e^{x} \) is \( \cos e^{x} \cdot e^x \).

Step 4: Combine the Two Derivatives

Add the derivatives of each part together to get the final derivative of the whole function: \[ \frac{d}{dx}(e^{\sin x} + \sin e^{x}) = e^{\sin x} \cos x + \cos e^{x} \cdot e^x \].

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. If you have a function composed of two other functions, say \( f(g(x)) \), the chain rule allows you to find its derivative. The rule states that to differentiate \( f(g(x)) \), you need to differentiate the outer function \( f \) with respect to \( g(x) \) and then multiply by the derivative of the inner function \( g \) with respect to \( x \).

• For example, if you have a function \( h(x) = e^{\sin x} \), apply the chain rule by setting \( u = \sin x \). First, differentiate \( e^u \) to get \( e^u \), then multiply by the derivative of \( \sin x \), which is \( \cos x \). This gives you the derivative as \( e^{\sin x} \cos x \).
• In the function \( \sin e^x \), set \( u = e^x \). Differentiate \( \sin u \) to get \( \cos u \), then multiply by the derivative of \( e^x \), which is \( e^x \), resulting in \( \cos e^x e^x \).

Differentiation

Differentiation is the process of finding the derivative of a function. It measures the rate at which a function changes at any given point. In mathematics, this is represented as \( \frac{dy}{dx} \) or \( f'(x) \). Derivatives also describe the slope of a tangent line to the curve of a function at a given point, providing insight into its behavior, such as increasing or decreasing trends.

• When you differentiate a function with respect to \( x \), you determine how the function's output changes with slight changes in input \( x \). For example, finding the derivative of \( e^{\sin x} + \sin e^x \) involves applying rules of differentiation to each component.
• It's essential for solving practical problems in physics, engineering, and economics, among others, as it helps understand the motion, growth, or change within a system.

Exponential Functions

Exponential functions are defined as mathematical functions of the form \( f(x) = a^{x} \), where \( a \) is a constant base. The most notable exponential function involves the constant \( e \), where \( e \approx 2.71828 \).
When differentiating exponential functions such as \( e^{u(x)} \), use the chain rule: the derivative is \( e^u \cdot \frac{du}{dx} \).

• In the example \( e^{\sin x} \), the exponent \( \sin x \) has its derivative \( \cos x \), leading to the derivative \( e^{\sin x} \cos x \).
• For \( e^x \), the derivative is straightforward, \( e^x \), because the rate of change of \( e^x \) is itself, simplifying many calculations.

Exponential growth and decay are common in real-world applications, illustrating rapid increases or decreases, such as in population growth or radioactive decay.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the lengths of its sides. These functions are periodic and play a vital role in modeling wave-like phenomena.
When differentiating trigonometric functions, remember:

• The derivative of \( \sin x \) is \( \cos x \), which means as the sine function increases, its rate of change is given by the cosine function.
• The derivative of \( \cos x \) is \(-\sin x \) due to its downward slope at the peaks of \( \cos x \).

Using these derivatives and the chain rule, you can differentiate composite trigonometric functions like \( \sin e^x \). First, find \( \cos(e^x) \), then multiply by the derivative of \( e^x \), leading to \( \cos e^x \cdot e^x \).
These derivatives are crucial for solving problems involving oscillations and waves, common in physics and engineering.