Match List-I with List-II : Choose the correct answer from the options given below :List-I
FunctionList-II
Derivative w.r.t. x(A) \( \frac{5^x}{\log_e 5} \) (I) \( 5^x (\log_e 5)^2 \) (B) \( \log_e 5 \) (II) \( 5^x \log_e 5 \) (C) \( 5^x \log_e 5 \) (III) \( 5^x \) (D) \( 5^x \) (IV) 0
4. (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
Understanding Derivatives and Functions Matching
This problem requires us to calculate the derivative of each function given in List-I with respect to \(x\) and then match it with the corresponding expression in List-II. Let's go through each function and find its derivative.
Calculating Derivatives for List-I Functions
Function (A): \( f(x) = \frac{5^x}{\log_e 5} \)
Here, \( \log_e 5 \) is a constant. We can write \( f(x) \) as \( f(x) = \frac{1}{\log_e 5} \cdot 5^x \). The derivative of \( a^x \) with respect to \(x\) is \( a^x \log_e a \).
So, the derivative of \( f(x) \) is:
\( \frac{d}{dx} \left( \frac{5^x}{\log_e 5} \right) = \frac{1}{\log_e 5} \frac{d}{dx} (5^x) \)
\( = \frac{1}{\log_e 5} (5^x \log_e 5) \)
\( = 5^x \)
This matches expression (III) in List-II.
Function (B): \( f(x) = \log_e 5 \)
Here, \( \log_e 5 \) is a constant value, approximately 1.609. The derivative of any constant with respect to \(x\) is always 0.
\( \frac{d}{dx} (\log_e 5) = 0 \)
This matches expression (IV) in List-II.
Function (C): \( f(x) = 5^x \log_e 5 \)
Here, \( \log_e 5 \) is a constant. We can write \( f(x) \) as \( f(x) = \log_e 5 \cdot 5^x \). The derivative of \( a^x \) is \( a^x \log_e a \).
So, the derivative of \( f(x) \) is:
\( \frac{d}{dx} (5^x \log_e 5) = \log_e 5 \frac{d}{dx} (5^x) \)
\( = \log_e 5 (5^x \log_e 5) \)
\( = 5^x (\log_e 5)^2 \)
This matches expression (I) in List-II.
Function (D): \( f(x) = 5^x \)
This is a direct application of the derivative formula for \( a^x \), where \( a = 5 \).
\( \frac{d}{dx} (5^x) = 5^x \log_e 5 \)
This matches expression (II) in List-II.
Summary of Matches
Based on our calculations, the matches are:
- (A) \( \frac{5^x}{\log_e 5} \) matches (III) \( 5^x \)
- (B) \( \log_e 5 \) matches (IV) \( 0 \)
- (C) \( 5^x \log_e 5 \) matches (I) \( 5^x (\log_e 5)^2 \)
- (D) \( 5^x \) matches (II) \( 5^x \log_e 5 \)
We can represent this in a table:
| List-I Function | List-II Derivative | Match |
|---|---|---|
| (A) \( \frac{5^x}{\log_e 5} \) | (III) \( 5^x \) | (A) - (III) |
| (B) \( \log_e 5 \) | (IV) \( 0 \) | (B) - (IV) |
| (C) \( 5^x \log_e 5 \) | (I) \( 5^x (\log_e 5)^2 \) | (C) - (I) |
| (D) \( 5^x \) | (II) \( 5^x \log_e 5 \) | (D) - (II) |
Checking the given options, the correct combination is (A) - (III), (B) - (IV), (C) - (I), (D) - (II).
Revision Table: Common Derivatives
| Function \(f(x)\) | Derivative \(f'(x)\) | Notes |
|---|---|---|
| \(c\) (constant) | \(0\) | Derivative of any constant is zero. |
| \(x^n\) | \(nx^{n-1}\) | Power rule. |
| \(e^x\) | \(e^x\) | Exponential function base \(e\). |
| \(a^x\) | \(a^x \log_e a\) | Exponential function base \(a > 0, a \neq 1\). |
| \( \log_e x \) | \( \frac{1}{x} \) | Natural logarithm. |
| \( \log_a x \) | \( \frac{1}{x \log_e a} \) | Logarithm with base \(a\). |
Additional Information on Derivatives and Calculus
Derivatives are a fundamental concept in calculus that measure the instantaneous rate of change of a function. They represent the slope of the tangent line to the graph of the function at a specific point.
- Constant Rule: The derivative of a constant function is always zero because a constant function's value does not change as the input changes, meaning its rate of change is zero.
- Constant Multiple Rule: If \(c\) is a constant and \(f(x)\) is a differentiable function, then \( \frac{d}{dx} [c \cdot f(x)] = c \cdot \frac{d}{dx} [f(x)] \). This was applied when taking the derivative of \( \frac{5^x}{\log_e 5} \) and \( 5^x \log_e 5 \), treating \( \frac{1}{\log_e 5} \) and \( \log_e 5 \) as constants, respectively.
- Exponential Function Derivative: The derivative of \( a^x \) being \( a^x \log_e a \) is a standard result derived from the limit definition of the derivative or from the properties of logarithms and chain rule (writing \( a^x = e^{x \log_e a} \)).
Understanding these basic rules is crucial for solving problems involving derivatives and matching functions to their rates of change.
Important Questions from Continuity and Differentiability
If f(x) is defined as:
\[ f(x) = \begin{cases} kx + 1 & \text{if } x \le \pi \\ \cos x & \text{if } x > \pi \end{cases} \]
is continuous at x = π, then the value of k is:
Differentiation of \( \log_5 (\log x^2) \) w.r.t. \( x \) is
The sum of values of \( a \) and \( b \) such that the function \( f(x) \) defined by
\[ f(x) = \begin{cases} 3, & x \leq 1 \\ ax + b, & 1 < x < 5 \\ 10, & x \geq 5 \end{cases} \] is a continuous function is
If \( f(x) = \begin{cases} \frac{\tan (\frac{\pi}{4} - x)}{\cot 2x}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases} \) is continuous at \( x = \frac{\pi}{4} \), then the value of \( k \) will be equal to:
If \( y = \frac{e^{-x} + e^x}{e^{-x} - e^x} \), then \( \frac{dy}{dx} \) is equal to:
