Derivatives of exponential and logarithmic functions from "summary" of Differential Calculus by S Balachandra Rao
When we consider exponential functions, we are looking at functions of the form \(y = a^x\), where \(a\) is a positive constant. These functions have a unique property - the rate of change of the function at any point is proportional to the value of the function at that point. In other words, the derivative of an exponential function is proportional to the function itself. This fundamental property makes derivatives of exponential functions particularly interesting and important in calculus. To find the derivative of an exponential function, we can use the definition of the derivative and the properties of exponential functions. By differentiating \(y = a^x\) with respect to \(x\), we obtain the derivative \(dy/dx = a^x \ln(a)\). This derivative shows us that the rate of change of an exponential function is proportional to the function itself, with the constant of proportionality being \(\ln(a)\), where \(\ln\) denotes the natural logarithm. Moving on to logarithmic functions, we consider functions of the form \(y = \log_a(x)\), where \(a\) is the base of the logarithm. Logarithmic functions are the inverse of exponential functions, and as such, their derivatives exhibit certain properties that mirror those of exponential functions. The derivative of a logarithmic function \(y = \log_a(x)\) is given by \(dy/dx = 1/(x \ln(a))\). This derivative shows us that the rate of change of a logarithmic function is inversely proportional to the input value \(x\), with the constant of proportionality being \(1/\ln(a)\). It is worth noting that both exponential and logarithmic functions play significant roles in various fields of mathematics and science. Their derivatives are essential in calculus for analyzing rates of change, optimization problems, and many other applications. Understanding the properties and behaviors of exponential and logarithmic functions, as well as their derivatives, is crucial for mastering differential calculus and its practical implications.
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