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...
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