The Chain Rule | Finding the Derivative of ln(x) with Respect to x

dy/dx ln x

To find the derivative of ln(x) with respect to x (denoted as dy/dx), we can use the chain rule

To find the derivative of ln(x) with respect to x (denoted as dy/dx), we can use the chain rule.

The chain rule states that if we have a composite function u(v(x)), then the derivative of u with respect to x is given by the product of the derivative of u with respect to v, and the derivative of v with respect to x.

In this case, our composite function is ln(x), where u(v) = ln(v) and v(x) = x. So, we need to find the derivative of u with respect to v and the derivative of v with respect to x.

The derivative of u with respect to v can be easily found. The derivative of ln(v) with respect to v is 1/v.

Next, we need to find the derivative of v with respect to x. In this case, v(x) = x, so the derivative of v with respect to x is 1.

Now, we can apply the chain rule. The derivative of ln(x) with respect to x is given by:

dy/dx = (1/v) * 1

Substituting v(x) = x, we have:

dy/dx = (1/x) * 1 = 1/x

Therefore, the derivative of ln(x) with respect to x is 1/x.

Definition:

ln(x), or natural logarithm of x, is a mathematical function that gives the logarithm of a positive real number x to the base of the mathematical constant e (approximately equal to 2.71828). It is the inverse function of the exponential function y = e^x. The natural logarithm is widely used in many areas of mathematics and science, including calculus, algebra, and statistics.

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