Question

Calculate the derivative of the following functions. $$y=\frac{1}{\log _{4} x}$$

Short answer

Answer: The derivative of the function \(y = \frac{1}{\log_4 x}\) with respect to x is \(\frac{dy}{dx} = \frac{-\ln{4}}{x(\ln{x})^2}\).

Step-by-step solution

Rewrite the function using natural logarithm

Before taking the derivative, rewrite the function using natural logarithm. Using the change of base formula for logarithms, which states that: $$\log_{b}a=\frac{\ln{a}}{\ln{b}}$$ The given function can be rewritten as: $$y=\frac{1}{\log _{4} x}=\frac{1}{(\frac{\ln{x}}{\ln{4}})}=\frac{\ln{4}}{\ln{x}}$$

Apply the chain rule and quotient rule

In order to find the derivative of the function, we need to apply the chain rule and the quotient rule simultaneously. The chain rule states that \((f(g(x)))' = f'(g(x))\cdot g'(x)\) while the quotient rule states that \((\frac{u}{v})' = \frac{u'v - uv'}{v^2}\). Let \(u = \ln{4}\) which is a constant, and \(v = \ln{x}\).

Calculate derivatives of u and v

Calculate the derivatives of u and v with respect to x: $$u' = \frac{d(\ln{4})}{dx} = 0 \quad (\text{since} \; \ln{4} \; \text{is a constant})$$ $$v' = \frac{d(\ln{x})}{dx} = \frac{1}{x}$$

Substitute the derivatives into the quotient rule formula

Now, substitute the calculated derivatives into the quotient rule formula: $$y'=\frac{u'v - uv'}{v^2}=\frac{(0)(\ln{x})-(\ln{4})(\frac{1}{x})}{(\ln{x})^2}=\frac{-\ln{4}\cdot\frac{1}{x}}{(\ln{x})^2}$$

Simplify the derivative

Finally, simplify the expression for the derivative: $$y'=\frac{-\ln{4}}{x(\ln{x})^2}$$ Thus, the derivative of the given function is: $$\frac{dy}{dx}=\frac{-\ln{4}}{x(\ln{x})^2}$$

Key concepts

Derivative

In mathematics, the derivative is a measure of how a function changes as its input changes. It's often visualized as the slope of the function at any given point. If you imagine a curve on a graph, the derivative at a particular point gives you the slope of the tangent line to the curve at that point. It's calculated using limits and provides allows us to understand the instantaneous rate of change of a function.
Derivatives have wide-ranging applications, from physics, where they can describe velocity and acceleration, to economics, for optimizing profit or cost functions.
When dealing with a problem in calculus like finding the derivative of a function such as \(y = \frac{1}{\log_{4} x}\), it's important to use rules and techniques that simplify the process. In more complex situations, we use additional concepts like the chain rule and the quotient rule to find derivatives.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a logarithm to the base of Euler's number \(e\), where \(e\) is approximately equal to 2.71828. It is widely used in calculus because it provides a natural and simple way to integrate and differentiate functions, especially when compared to other logarithmic bases.
Using the natural logarithm simplifies many calculus problems. In this example, converting the expression \(\log_{4}x\) using the base change formula to \(\frac{\ln x}{\ln 4}\) allows us to work with a simpler mathematical form, opening pathways to differentiation. This skill is crucial for rewriting expressions so they fit into known formulas and rules when performing operations like finding derivatives.

Chain Rule

The chain rule is a fundamental tool used for finding the derivative of composite functions. A composite function is a function made up of two or more simpler functions. The chain rule states:

• If you have a composite function \(f(g(x))\), the derivative \((f(g(x)))'\) is the product of \(f'\) of \(g(x)\) and \(g'(x)\).

This technique is particularly useful when functions nest within each other or depend on previous computations, a common scenario in real-world applications.
In the example problem, identifying the inner and outer functions and using the chain rule is essential. When handling logarithmic expressions converted to the natural logarithm, applying the chain rule ensures that we correctly differentiate nested functions and accurately account for each component part.

Quotient Rule

The quotient rule is another key tool in calculus, used to find the derivative of a function that is a ratio of two differentiable functions. It is expressed as:

• For \(u/v\), \(u\) and \(v\) being functions, the derivative \((\frac{u}{v})'\) is given by \(\frac{u'v - uv'}{v^2}\).

This rule is particularly important when dealing with fractions, as in our problem where the function \(y = \frac{\ln 4}{\ln x}\) needs to be differentiated.
The first step is to identify the numerator and the denominator functions. Then, calculate their derivatives separately. Finally, apply these results into the quotient rule formula to get the derivative of the given function. This systematic approach allows us to handle complex fractions effectively and helps to ensure that no steps are skipped during differentiation.