Problem 2 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(y=\log _{10} x^{2}\),then \(\frac{d y}{d x}\) is equal to (a) \(2 / x\) (b) \(\frac{2}{x \log _{e} 10}\) (c) \(\frac{1}{x \log _{e} 10}\) (d) \(\frac{1}{10 x}\)

Step-by-Step Solution

Answer

The correct answer is (b) \(\frac{2}{x \log_{e} 10}\).

Step 1: Identify the Function to Differentiate

We start with the function given in the problem: \(y = \log_{10} x^2\). This can be rewritten using logarithm properties as \(y = 2\log_{10} x\).

Step 2: Express Logarithm in Natural Logarithm

To differentiate the expression, it's often easier to work with natural logarithms. Recall the change of base formula \(\log_b a = \frac{\ln a}{\ln b}\). Apply this to express the equation as \(y = \frac{2\ln x}{\ln 10}\).

Step 3: Differentiate Using the Chain Rule

Differentiate \(y = \frac{2\ln x}{\ln 10}\) with respect to \(x\). The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\). Hence, \(\frac{dy}{dx} = \frac{2}{\ln 10} \cdot \frac{1}{x}\).

Step 4: Simplify the Derivative Expression

Simplify the expression to get \(\frac{dy}{dx} = \frac{2}{x \ln 10}\). This expression matches one of the options.

Key Concepts

Chain Rule in Calculus

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. Imagine that a function is made up of two or more functions combined together, one within the other. To differentiate such a function, we use the chain rule. It guides us on how to handle these layers of functions correctly.

To better grasp this idea, let’s consider a function where one function is nested inside another. For instance, if we have a composite function like \(z = f(g(x))\), to find its derivative \(\frac{dz}{dx}\), we need to use the chain rule. This involves:

• First, finding the derivative of the outer function \(f\), treating \(g(x)\) as an inside part. This gives us \(f'(g(x))\).
• Then, multiplying this by the derivative of the inner function \(g(x)\), which is \(g'(x)\).

So, the chain rule can be expressed as:\[ \frac{dz}{dx} = f'(g(x)) \cdot g'(x) \]This approach helps us tackle the derivative even when functions get complex or layered.

Change of Base Formula

Logarithms can be initially challenging, but the change of base formula is handy for simplifying expressions with logarithms. The change of base formula allows you to convert a logarithm from one base to another. This is vital when differentiating logarithmic functions, especially when the base isn't the natural logarithm \(\ln\).

The formula is stated as:\[ \log_b a = \frac{\ln a}{\ln b} \]It converts a logarithm with base \(b\) into natural logarithms, using the well-known properties of \(\ln\). For instance, if you have \(\log_{10} x\), you can express it as \(\frac{\ln x}{\ln 10}\).

By using the change of base formula, you can differentiate any logarithmic function using the calculus techniques that apply to natural logs. This trick not only makes differentiation easier but also unifies different logarithmic bases into a consistent method.

Derivative of Logarithmic Functions

Differentiating logarithmic functions is often a simpler task when approached correctly. The key is to know the basic derivative of the natural logarithm function. The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\). This foundation allows us to handle more complicated logarithmic functions.

When you’ve expressed your logarithmic function using the natural log (thanks to the change of base formula), you can straightforwardly find its derivative. For example, if you have a function like \(y = \ln x\), the derivative \( \frac{dy}{dx} = \frac{1}{x} \).

• Recommending expressing non-natural base logs in terms of \(\ln x\).
• Differentiate using \(\frac{1}{x}\), modifying the expression with any coefficients or constants involved as necessary.

This method ensures you smoothly navigate through any differentiation involving logs, no matter how complex they get.