Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (4x+1)^{In x}

검증된 단계별 안내

Step 1: To find the derivative using the fact that b^x = e^(x ln b), rewrite the function y = (4x+1) ln x in terms of exponential functions. This involves expressing any power terms in the form of e^(x ln b).

Step 2: Apply the product rule to differentiate y = (4x+1) ln x. The product rule states that if you have a function y = u*v, then the derivative y' = u'v + uv'. Here, u = (4x+1) and v = ln x.

Step 3: Differentiate u = (4x+1) to get u' = 4. Differentiate v = ln x to get v' = 1/x. Substitute these derivatives into the product rule formula to find the derivative of y.

Step 4: For logarithmic differentiation, take the natural logarithm of both sides of the equation y = (4x+1) ln x, which gives ln y = ln((4x+1) ln x). Differentiate both sides with respect to x using implicit differentiation.

Step 5: After differentiating, solve for dy/dx. Compare the derivative obtained from logarithmic differentiation with the derivative obtained using the product rule to verify that both methods yield the same result.

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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 can be calculated using various rules, such as the power rule, product rule, and chain rule, depending on the form of the function.

Exponential and Logarithmic Functions

Exponential functions are of the form b^x, where b is a constant and x is the variable. They can be expressed using natural logarithms as b^x = e^(x ln b). Logarithmic functions, the inverses of exponential functions, help simplify the differentiation of complex expressions, especially when dealing with products or powers, making them essential for techniques like logarithmic differentiation.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or powers of variables. By taking the natural logarithm of both sides of the equation, the differentiation process becomes simpler, allowing the use of properties of logarithms to break down complex expressions. This method is particularly useful for functions where the variable is in both the base and the exponent.

Calculate the derivative of the following functions (i) using the fact that bx = exIn b and (ii) using logarithmic differentiation. Verify that both answers are the same.y = (4x+1)In x