Question

Compute the derivative of the given function. $$f(x)=\ln \left(x^{2}\right)$$

Short answer

The derivative is \( f'(x) = \frac{2}{x} \).

Step-by-step solution

Identify the function and rewrite it

The given function is \( f(x) = \ln(x^2) \). Using the property of logarithms \( \ln(a^b) = b \ln(a) \), we can rewrite the function as \( f(x) = 2 \ln(x) \).

Apply the derivative rule for logarithmic functions

We need to find the derivative of \( f(x) = 2 \ln(x) \). The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \).

Differentiate the function

Differentiate \( f(x) = 2 \ln(x) \). Applying the constant multiple rule (derivative of \( c \cdot g(x) \) is \( c \cdot g'(x) \)), we get:\[ f'(x) = 2 \cdot \frac{1}{x} = \frac{2}{x}. \]

Key concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique in calculus used primarily for functions that are products, quotients, or powers of variables. This method is particularly useful when differentiating complex expressions where applying standard differentiation rules directly could be cumbersome. The process generally involves taking the natural logarithm of both sides of the equation and then differentiating implicitly.

• Start by introducing a logarithm to simplify the algebraic expression.
• Utilize properties of logarithms, such as \( \ln(a^b) = b \ln(a) \), to break down the expression.
• Differentiate the resulting expression using standard differentiation rules.

Ultimately, this approach transforms difficult multiplication or exponentiation operations into manageable addition or multiplication scenarios. It's especially useful for functions where the power is another function itself. This can simplify the differentiation process, making it more straightforward to find the derivative of otherwise complex functions.

Power Rule

The power rule is one of the fundamental rules for differentiation, commonly used due to its simplicity and wide applicability. It's used to differentiate functions of the form \( x^n \), where \( n \) is any real number. The rule states that if \( f(x) = x^n \), then the derivative, denoted \( f'(x) \), is given by \( f'(x) = nx^{n-1} \).

• This means you multiply the function by its exponent.
• Subtract one from the original exponent to get the new exponent.

For instance, if given a function \( f(x) = x^3 \), the derivative would be \( f'(x) = 3x^2 \).
The power rule is integral to quickly finding derivatives and forms the basis for many other more complex differentiation techniques. Its straightforward application makes it an invaluable tool in calculus, providing an easy way to handle polynomial terms in functions.

Constant Multiple Rule

The constant multiple rule in differentiation is a simple yet essential concept. This rule helps simplify the process of taking derivatives when the function is multiplied by a constant. The rule states that if you have a function \( f(x) = c \cdot g(x) \), where \( c \) is a constant, then the derivative \( f'(x) \) is \( c \cdot g'(x) \).

• This allows the constant to be "moved outside" the differentiation process.
• Apply the differentiation to the remaining function as usual.

For example, if \( f(x) = 3x^2 \), the derivative will be \( f'(x) = 3 \cdot 2x = 6x \).
Using this rule simplifies differentiation by effectively reducing it to a single step of focusing on the non-constant part of the function. By extracting constants, it becomes easier to handle complex expressions and apply other differentiation rules.

Chain Rule

The chain rule is an invaluable tool in calculus used for finding derivatives of compositions of functions. When a function is nested within another, such as \( f(g(x)) \), the chain rule provides a way to differentiate this composite function. The basic formula for the chain rule is:\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \].

• First, identify the inner function \( g(x) \) and the outer function \( f(x) \).
• Differentiate the outer function while keeping the inner function unchanged, then multiply by the derivative of the inner function.

For example, given a function \( h(x) = (3x+2)^4 \), you identify \( g(x) = 3x+2 \) and \( f(u) = u^4 \).
By applying the chain rule:

• Differentiate \( f(u) \) to get \( 4u^3 \).
• Substitute back \( g(x) = 3x+2 \).
• Multiply by the derivative of \( g(x) \), which is 3, resulting in \( h'(x) = 4(3x+2)^3 \cdot 3 \).

This rule is critical for efficiently handling complex, layered functions and is frequently used alongside other differentiation rules for a comprehensive analysis of a function's derivative.