06_Ex 2.6

Exercise Questions

Questions	Question Links
Q1. Find $f^{\prime}(\boldsymbol{x})$ if…	2.6 Q-1
Q2. Find $\frac{d y}{d x}$ if…	2.6 Q-2
Q3. Find $\frac{d y}{d x}$ if…	2.6 Q-3

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Overview

This exercise focuses on the derivatives of exponential, hyperbolic, and logarithmic functions. The application of these derivatives is essential in calculus, particularly for solving real-world problems involving growth, decay, and oscillations. By mastering these derivatives, students will be able to tackle more complex differential equations and functions encountered in physics, engineering, and other fields.

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Key Concepts

1. Exponential Functions: A function of the form $f(x) = a^x$ where $a > 0$ and $a \neq 1$. When $a = e$the function is called the natural exponential function, $f(x) = e^x$.
2. Hyperbolic Functions: These are analogs of trigonometric functions but based on exponential functions, such as:
   • $\sinh x = \frac{e^x - e^{-x}}{2}$
   • $\cosh x = \frac{e^x + e^{-x}}{2}$
3. Logarithmic Functions: A logarithmic function is the inverse of an exponential function. The natural logarithmic function is written as $\ln x$, and the common logarithmic function is written as $\log_a x$.

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Important Formulas

1. Derivative of Exponential Functions: $\frac{d}{dx} e^x = e^x$ $\frac{d}{dx} a^x = a^x \ln a$
2. Derivative of Hyperbolic Functions:
   • $\frac{d}{dx} \sinh x = \cosh x$
   • $\frac{d}{dx} \cosh x = \sinh x$
   • $\frac{d}{dx} \tanh x = 1 - \tanh^2 x$
3. Derivative of Logarithmic Functions:
   • $\frac{d}{dx} \ln x = \frac{1}{x}$
   • $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$

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Tips and Tricks

• For derivatives involving exponential functions, remember that $\frac{d}{dx} e^x = e^x$, which simplifies the calculation.

• When dealing with logarithmic derivatives, apply the chain rule carefully, especially for composite functions.

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Summary

In this exercise, we explored the derivatives of exponential, hyperbolic, and logarithmic functions. These derivatives are fundamental in calculus, with direct applications in many areas of science and engineering. Mastering them enables you to solve a wide variety of problems, including those involving growth and decay, oscillations, and other physical phenomena.

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Reference

By Great Science Academy:

By Sir Shahzad Sair: