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Find the derivative of $\ln\left(5x\right)$ 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 $\ln\left(5x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{\ln\left(5\left(x+h\right)\right)-\ln\left(5x\right)}{h}\right)$
Learn how to solve problems step by step online. Find the derivative of ln(5x) using the definition. Find the derivative of \ln\left(5x\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(5x\right). Substituting f(x+h) and f(x) on the limit, we get. Solve the product 5\left(x+h\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). Factor the polynomial 5x+5h by it's greatest common factor (GCF): 5.