Find the derivative of the following functions.
y = In(e^x + e^-x)

검증된 단계별 안내

Step 1: Recognize that the function y = ln(e^x + e^-x) is a composition of functions, where the outer function is the natural logarithm ln(u) and the inner function is u = e^x + e^-x.

Step 2: Apply the chain rule for differentiation, which states that if y = ln(u), then the derivative dy/dx = (1/u) * (du/dx).

Step 3: Differentiate the inner function u = e^x + e^-x with respect to x. The derivative of e^x is e^x, and the derivative of e^-x is -e^-x.

Step 4: Substitute the derivative of the inner function back into the chain rule formula. This gives dy/dx = (1/(e^x + e^-x)) * (e^x - e^-x).

Step 5: Simplify the expression if possible. The derivative of y with respect to x is dy/dx = (e^x - e^-x) / (e^x + e^-x).

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주요 개념

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Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx, and it can be calculated using various rules such as the power rule, product rule, and chain rule.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is the inverse function of the exponential function e^x. Understanding the properties of natural logarithms, such as ln(a*b) = ln(a) + ln(b) and ln(a^b) = b*ln(a), is essential for differentiating functions that involve logarithmic expressions.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where a and b are constants, and e is the base of the natural logarithm. These functions exhibit rapid growth or decay and are characterized by their unique property that the derivative of e^x is e^x. In the context of the given function, recognizing how to differentiate expressions involving e^x and e^(-x) is crucial for finding the derivative.

Find the derivative of the following functions.y = In(e^x + e^-x)