Question

How do I find the derivative of $\ln (5x)$?

Answer 1

This question is from the topic of derivatives. In this question we need to find the derivative of function natural logarithm $\ln (5x)$. To solve this question requires knowledge of how to differentiate a function and chain rule of differentiation. To solve this we apply chain rule of differentiation to $\ln (5x)$.

Complete step by step solution:
Let us try to solve this question in which we are asked to find the derivative of natural logarithmic function $\ln (5x)$.
Before differentiating this, let’s have a look at definition of chain rule, suppose a function $f(x) = g(h(x))$ such that both $g$ and $h$ are differentiable with respect to $x$ then $f$ is also differentiable and its differentiation is given by $f'(x) = g'(h(x)) \cdot h'(x)$ where $f'(x) = \dfrac{{d(f(x))}}{{dx}}$ and similarly $g'$ and $h'$ are derivatives of functions $g$ and $h$ respectively.
Now, let’s find the derivative of function $\ln (5x)$. Function to derivative $\ln (5x)$ is composition of differentiable function $\ln (x)$ and $5x$.
So for the derivative of the function $\ln (5x)$ we can use the chain rule of differentiate.
After applying chain rule to function $\ln (5x)$, we get
$\dfrac{{d(\ln (5x))}}{{dx}} = \dfrac{{d(\ln (5x))}}{{dx}} \cdot \dfrac{{d(5x)}}{{dx}}$
$eq(1)$
As we know that $\dfrac{{d(\ln x)}}{{dx}} = \dfrac{1}{x}$. So we have,
$\dfrac{{d(\ln 5x)}}{{dx}} = \dfrac{1}{{5x}}$ $eq(2)$
And, also we know that $\dfrac{{d(a{x^n})}}{{dx}} = (na){x^{n - 1}}$. So we have,
$\dfrac{{d(5x)}}{{dx}} = 5$ $eq(3)$
Now, putting back the value of $eq(2)$ and $eq(3)$ in $eq(1)$, we get the derivative of $\ln (5x)$.
Hence the derivative of function $\dfrac{{d(\ln (5x))}}{{dx}} = \dfrac{1}{{5x}} \cdot 5 = \dfrac{1}{x}$.

Note: To solve these types of questions in which we are asked to find the derivative of a given function. For solving this type of question we are required to have knowledge of how to find derivatives of a function, differentiability of common function and properties of differentiation such sum rule, product rule, division rule and chain rule.