Find the derivative of each function. f(x)=ln(x e^√(x)+2) |...

Name: Problem 42, Chapter 12: Finite Mathematics and Calculus with Applications
Uploaded: 2020-05-12T00:14:33-08:00
Duration: 5 min 50 s
Channel: Oswaldo Jiménez
Description: Find the derivative of each function. f(x)=ln(x e^√(x)+2)

Key Concepts

Differentiation of Square Root Functions

This concept involves finding the derivative of a square root function. The function sqrt(x) can be rewritten as x^(1/2), and its derivative is computed using the power rule, resulting in 1/(2sqrt(x)).

Differentiation of Exponential Functions

Differentiation of exponential functions involves applying the chain rule when the exponent is a function of x. For example, for e^(sqrt(x)), differentiate the outer exponential function and then multiply by the derivative of the inner function, sqrt(x).

Product Rule

The product rule is essential when differentiating functions that are the product of two or more functions. In this context, when differentiating a product such as x · e^(sqrt(x)), you differentiate each factor separately and combine them, ensuring that the interaction between functions is properly accounted for.

Chain Rule

The chain rule is used to differentiate composite functions. When a function is applied to another function, like ln(g(x)) where g(x) is itself a composite function, the derivative is found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Logarithmic Differentiation

This concept involves taking the derivative of a logarithm of a function. Specifically, when you have a function in the form ln(g(x)), its derivative is g'(x)/g(x). This rule simplifies the process by allowing the derivative of an inside complicated function to be handled separately.

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