Derivatives are the basics of calculus and geometry and are used rigorously by direct or indirect ways in real life problems or scenarios. The derivative of a function is defined by the change in the function value with respect to unit change of the variable/s of the function. The derivative of the function can be calculated analytically as well as geometrically depending on its usage.
The derivative of ln(x) or the natural logarithm of x can be calculated by the following method.
y = f(x) = ln(x).
Hence, by the definition of derivative
Now, by standard limit theorem,
In a slightly modified theorem
Derivative is applied in many problems in mathematics and physics that includes optimization, related rates, Newton’s method, linear approximations and the mean value theorem.
Optimization: Many problems requires to find the maximum or minimum value of function and at which point those values are obtained. For example, it may be required to find the minimum time required to complete a journey or maximum power that can be generated by a device. This can be easily obtained if the expression of the function in terms of its variable is known. If the function is minimum at a particular point then the derivative of that function will be zero and double derivative at that point is positive. Conversely, the maximum of a function indicates that the derivative of that function is zero at that point and the double derivative at that point is negative. Hence, by using derivatives if there exists a maximum or minimum or both of the function can be found. However, derivatives can give the value of local maxima or minima and choosing suitable interval point is required to find the global maximum or minimum.
Related rates: If there are variables which are changing with respect to time then the related rate problem is defined by a problem where the change of the variable is needed to be found at a particular time instant. For example, two cars travelling given by velocity y = 3t^2 + 18t and z = 6t^3 + 14t.
Hence, the rate of change of position of first car
Now, the rate of change of position of second car
Hence, the relative rate of change of position of two cars or relative velocity change in t = 1 sec will be
Newton’s method: Newton’s method is an iterative root finding method where the poles of function is found using Newton’s formula. The formula involves calculating derivative of the given function which is given by,
Where xn, = current iteration point and xn+1 = next iteration point.
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