Final Answer
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Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(\sqrt{5x+1}\right)$ and $g=\left(x^3+4\right)^6$
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$\frac{d}{dx}\left(\ln\left(\sqrt{5x+1}\right)\right)\left(x^3+4\right)^6+\frac{d}{dx}\left(\left(x^3+4\right)^6\right)\ln\left(\sqrt{5x+1}\right)$
Learn how to solve differential equations problems step by step online. Find the derivative of ln((5x+1)^1/2)(x^3+4)^6. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\ln\left(\sqrt{5x+1}\right) and g=\left(x^3+4\right)^6. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}.