75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^10x

Verified step by step guidance

1. Step 1: Start by taking the natural logarithm of both sides of the equation y = f(x) = x^(10x). This gives us ln(y) = ln(x^(10x)).
2. Step 2: Use the property of logarithms that allows you to bring the exponent down: ln(y) = 10x * ln(x).
3. Step 3: Differentiate both sides with respect to x. On the left side, use implicit differentiation: (1/y) * (dy/dx). On the right side, apply the product rule to differentiate 10x * ln(x).
4. Step 4: The product rule states that if u(x) and v(x) are functions of x, then the derivative of their product is u'(x)v(x) + u(x)v'(x). Here, let u(x) = 10x and v(x) = ln(x). Differentiate to find u'(x) = 10 and v'(x) = 1/x.
5. Step 5: Substitute the derivatives back into the equation from Step 3: (1/y) * (dy/dx) = 10 * ln(x) + 10x * (1/x). Solve for dy/dx by multiplying both sides by y, and then substitute y = x^(10x) back into the equation to express dy/dx in terms of x.

Key Concepts

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 function, we can simplify the differentiation process, especially when dealing with complex expressions. This method is particularly useful for functions like f(x) = x^(10x), where both the base and the exponent are variable.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y = g(u) is composed with u = f(x), then the derivative dy/dx can be found by multiplying the derivative of g with respect to u by the derivative of f with respect to x. This rule is essential when applying logarithmic differentiation, as it allows us to differentiate the logarithm of a function effectively.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if u(x) and v(x) are two differentiable functions, then the derivative of their product is given by u'v + uv'. This rule is particularly relevant in logarithmic differentiation when we encounter products of functions, as it helps in breaking down the differentiation process into manageable parts.