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Find the derivative of $\ln\left(\frac{x}{y}\right)$ 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 $\ln\left(\frac{x}{y}\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{\ln\left(\frac{x+h}{y}\right)-\ln\left(\frac{x}{y}\right)}{h}\right)$
Learn how to solve definition of derivative problems step by step online. Find the derivative of ln(x/y) using the definition. Find the derivative of \ln\left(\frac{x}{y}\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 \ln\left(\frac{x}{y}\right). Substituting f(x+h) and f(x) on the limit, we get. 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). We can simplify the quotient of fractions \frac{\frac{x+h}{y}}{\frac{x}{y}} by inverting the second fraction and multiply both fractions. Simplify the fraction .