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How to Find Derivatives of Logarithms

Taking the derivative of functions with logarithms in them is an important skill you learn in calculus. The logarithm of a number with a specific base is the exponent you raise the base to to make that number. For example, the logarithm of 100 to base 10 is 2, because 10^2 is 100. There are two simple rules to help you solve the derivatives of log b(x) and the special case of ln(x).

Instructions

    • 1

      Use the rule stating that the derivative of log b(x) is 1/(x * ln(b)). For example, the derivative of log 5(x) is 1/(x * ln(5)).

    • 2

      Use the rule that the derivative of ln(x) is 1/x. For example, the derivative of ln(7) = 1/7.

    • 3

      Use the product rule to help solve more complicated derivatives. The general form of the product rule is h(x) = f(x)g(x) = f(x)g(x) + f(x)g(x). This is useful when you have a function that is the product of two other functions. You take the derivative of the first part multiplied by the unchanged second part then add it to the derivative of the second part multiplied by the unchanged first part. For example, the derivative of x * ln(x) becomes 1 * ln(x) + x * (1/x), which is ln(x) + 1.

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