Equations that involve variables raised to variable-based powers and other algebraic complexities can be difficult to differentiate because they follow different rules than standard equations. In these cases, you can use logarithmic differentiation in order to find the derivative. Logarithmic differentiation typically requires that you take the natural logarithm, or "ln," of both sides of the equations. This allows you to algebraically manipulate and simplify the equation so that it can be differentiated more easily.
Properties of Logarithms
The properties of logarithms allow you to separate elements of complicated functions. For example, if a function contains the product of two variable terms, you can separate them through the following property: ln(ab)=ln(a) + ln(b). Similarly, the logarithm of a quotient is found through subtraction: ln(a/b)=ln(a) - ln(b). Logarithms can also help you disentangle complicated exponential functions based on this property: ln(a^b) = b * ln(a). The more you practice with these properties, the more you will understand how much simpler they can make your calculus problems.
Important Calculus Rules
When you take the derivative of a function, you must utilize special rules, such as the product rule and chain rule. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the derivative of the second function times the first: [f(x)g(x)]' = f'(x)g(x) + g'(x)f(x). The chain rule states that the derivative of a nested function is equal to the derivative of the first function times the derivative of the second function: [f(g(x))]' = f'(g(x))g'(x). The chain rule is especially important because once you have rearranged your equation, the left-hand side will be a nested function; it will look like this: ln(f(x)).
Derivative of the Natural Logarithm
The derivative of ln(x) is 1/x. This means that taking the natural logarithm of both sides of an equation will not only allow you to algebraically simplify the equation but will also result in a number of fairly simple derivatives. This is not the case with logarithms of any other base. While you could take the common logarithm of both sides of an equation, you would not be able to take the derivative of that equation as easily as you can with the natural logarithm.
Example of Logarithmic Differentiation
Suppose you have the function f(x) = 2x^(3x). You might be tempted to write that f'(x) = (3x)(2x^(3x-1)). However, this is incorrect. This is a situation in which you must use logarithmic differentiation. First, take the natural log of both sides: ln(f(x)) = ln(2x^(3x)). Then, use the properties of logarithms to separate the terms on the right side: ln(f(x)) = (3x)(ln(2x)). Then, take the derivative of both sides: ln(f(x))' = [(3x)(ln(2x))]'. On the left hand side, use the chain rule: ln(f(x))' = f'(x)/f(x). On the right hand side, use the product rule: [(3x)(ln(2x))]' = 3(ln(2x))+(3x)(2/2x)= 3(ln(2x))+3. Therefore, you have f'(x)/f(x) = 3(ln(2x))+3. You can multiply both sides by f(x) to isolate f'(x): f'(x) = (3(ln(2x))+3)(f(x)). Finally, substitute in f(x) = 2x^(3x): f'(x) = (3(ln(2x))+3)(2x^(3x)).
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