Find d y / d x . y=(x^2+1)^lnx

Name: Problem 31, Chapter 6: Calculus
Uploaded: 2021-06-16T16:42:52-08:00
Duration: 2 min 11 s
Channel: Rukhmani Jain
Description: Find d y / d x . y=(x^2+1)^lnx

step 1 In this question we have to differentiate the given function that is y equals to x square plus 1

Question

Find $d y / d x .$ $$y=\left(x^{2}+1\right)^{\ln x}$$

   Find $d y / d x .$
$$y=\left(x^{2}+1\right)^{\ln x}$$

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 6, Problem 31

Instant Answer

Solved by Expert Rukhmani Jain

06/16/2021

Step 1

This gives us: $$\ln y = \ln x \cdot \ln (x^2 + 1)$$ Show more…

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Find $d y / d x .$ $$y=\left(x^{2}+1\right)^{\ln x}$$

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Key Concepts

Logarithmic Differentiation

This is a method used when dealing with functions where both the base and the exponent are functions of the variable. By taking the natural logarithm of both sides of the equation, the exponent can be brought down, turning the original expression into a product that is easier to differentiate.

Chain Rule

The chain rule is used in differentiation when dealing with composite functions. In problems like this, after applying logarithmic differentiation, the chain rule helps in differentiating the inner functions, such as the logarithm of the base or the exponent.

Product Rule

When the differentiated expression results in a product of two functions, the product rule is used to differentiate the product. This is especially relevant when the logarithm of the original function yields a product of two separate functions that each depend on the variable.

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