75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+x²)^sin x

Verified step by step guidance

1. Step 1: Begin by taking the natural logarithm of both sides of the equation y = (1 + x²)^sin(x). This gives us ln(y) = ln((1 + x²)^sin(x)).
2. Step 2: Use the property of logarithms that allows you to bring the exponent down: ln(y) = sin(x) * ln(1 + x²).
3. Step 3: Differentiate both sides with respect to x. For the left side, use implicit differentiation: (1/y) * dy/dx. For the right side, use the product rule: d/dx[sin(x) * ln(1 + x²)].
4. Step 4: Apply the product rule to the right side: d/dx[sin(x)] * ln(1 + x²) + sin(x) * d/dx[ln(1 + x²)].
5. Step 5: Simplify the derivatives: cos(x) * ln(1 + x²) + sin(x) * (2x / (1 + x²)). Finally, solve for dy/dx by multiplying both sides by y, which is (1 + x²)^sin(x).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of other functions, especially when they involve exponentiation. By taking the natural logarithm of both sides of the function, we can simplify the differentiation process, particularly when the function is in the form of y = f(x) = g(x)^h(x). This method allows us to use properties of logarithms to bring down exponents and transform products into sums.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and v, such that y = f(u) and u = g(x), then the derivative of y with respect to x is the product of the derivative of f with respect to u and the derivative of g with respect to x. This rule is essential when dealing with functions that have nested structures, as seen in the given function f(x).

Implicit Differentiation

Implicit differentiation is a technique used when a function is not explicitly solved for one variable in terms of another. Instead of isolating y, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method is particularly useful in cases where the relationship between variables is complex, allowing us to find derivatives without needing to express one variable solely in terms of the other.