Derivative 6Lnx – What Is The Derivative Of 6Ln(X)

I need help differentiating. I am really confused how to solve with the $ln x$ in the equation. Which of the logarithm rules do I need to use for this equation?

$$y= 12x ln x + 12x – 6x (ln x)^2 + 8$$

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You wouldn”t use any of the normal logarithmic rules, you”d instead use the product and chain rule. Namely, you get $$frac{wnyrails.orgrm{d}y}{wnyrails.orgrm{d}x} = 12ln x + 12x cdot frac{1}{x} + 12 – 6 (ln x)^2 – 6x cdot 2 ln x cdot frac{1}{x}$$

This simplifies to $$frac{wnyrails.orgrm{d}y}{wnyrails.orgrm{d}x} = 24 – 6(ln x)^2$$

The product rule states that for a product of function $uv$, the derivative is given by $$frac{wnyrails.orgrm{d}}{wnyrails.orgrm{d}x} uv = u , wnyrails.orgrm{d}v + v, wnyrails.orgrm{d}u$$

The chain rule states that for a composition of function $f(g(x))$, the derivative is given by $$frac{wnyrails.orgrm{d}}{wnyrails.orgrm{d}x} f(g(x)) = g”(x)f”(g(x))$$

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To find the derivative of $12xln x$, use the product rule to get$$12ln x + 12x(1/x) = 12(ln x +1)$$

To find the derivative of $(6x)(ln x)^2$, use the product rule and the chain rule to get$$6(ln x)^2 + 6x(2)(ln x)(1/x) = 6ln x((ln x) + 2)$$

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