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 = (x²+1)^x

Verified step by step guidance

1. Step 1: For part (i), rewrite the function y = (x²+1)^x using the property b^x = e^(x ln b). This gives y = e^(x ln(x²+1)).
2. Step 2: Differentiate y = e^(x ln(x²+1)) with respect to x using the chain rule. The derivative of e^u with respect to x is e^u * du/dx, where u = x ln(x²+1).
3. Step 3: Find du/dx where u = x ln(x²+1). Use the product rule for differentiation: if u = v * w, then du/dx = v' * w + v * w'. Here, v = x and w = ln(x²+1).
4. Step 4: For part (ii), use logarithmic differentiation. Take the natural logarithm of both sides: ln(y) = x ln(x²+1). Differentiate both sides with respect to x. The left side becomes (1/y) * dy/dx, and the right side is differentiated using the product rule.
5. Step 5: Solve for dy/dx in both methods and verify that the expressions obtained are equivalent, confirming that both methods yield the same derivative.

Key Concepts

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 and techniques, such as the power rule, product rule, and chain rule.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = b^x, where b is a positive constant. They exhibit rapid growth or decay and are characterized by their unique property that the rate of change of the function is proportional to its current value. The expression b^x can be rewritten using the natural exponential function as e^(x ln b), which simplifies differentiation.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to variable powers. By taking the natural logarithm of both sides of the equation, the differentiation process becomes simpler, allowing the use of properties of logarithms to separate variables. This method is particularly useful for functions like y = (x² + 1)^x, where both the base and the exponent are variable.