Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function

Answer 1

Answer:  $\dfrac{x^2}{(x^3+1)}$

Step-by-step explanation

Properties of derivative , we use here :

• $\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}$
• $\dfrac{d}{dx}(x^n)=n(x)^{n-1}$
• $\dfrac{d}{dx}(a)=0$ , where a is constant.

The given function : $y=\ln(x^3+1)^{\frac{1}{3}}$

$y=\dfrac{1}{3}\cdot \ln (x^3+1)$  [$\because n\ln x=\ln x^n$]

Now , Differentiate both sides , we get

$\dfrac{dy}{dx}=\dfrac{1}{3}\cdot \dfrac{1}{(x^3+1)}\cdot \dfrac{d}{dx}(x^3+1)$

(By chain rule)

$=\dfrac{1}{3}\cdot \dfrac{1}{(x^3+1)}\cdot (3x^2+0)$

$=\dfrac{1}{3}\cdot \dfrac{1}{(x^3+1)}(3x^2)$

$=\dfrac{x^2}{(x^3+1)}$

Hence, the derivative of the function will be : $\dfrac{x^2}{(x^3+1)}$