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Find the derivative of $8\ln\left(x\right)$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $8\ln\left(x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{8\ln\left(x+h\right)-8\ln\left(x\right)}{h}\right)$
Learn how to solve problems step by step online. Find the derivative of 8ln(x) using the definition. Find the derivative of 8\ln\left(x\right) using the definition. Apply the definition of the derivative: \displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}. The function f(x) is the function we want to differentiate, which is 8\ln\left(x\right). Substituting f(x+h) and f(x) on the limit, we get. Factor the polynomial 8\ln\left(x+h\right)-8\ln\left(x\right) by it's greatest common factor (GCF): 8. The limit of the product of a function and a constant is equal to the limit of the function, times the constant: \displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}. The difference of two logarithms of equal base b is equal to the logarithm of the quotient: \log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right).