Question

How do you find the equation of the tangent line to the curve $y=\ln \left( 3x-5 \right)$ at the point where x=3?

Answer 1

In this question, we are going to find out the tangent line to the curve $y=\ln \left( 3x-5 \right)$ using the formula of differentiation. For finding the tangent of any curve, we have to differentiate the curve. That’s why here we are going to use the differentiating formula is $\dfrac{dy}{dx}=\dfrac{d\left( \ln x \right)}{dx}$ which will be equal to$\dfrac{1}{x}$.
After using differentiation formulas or rules, we will apply here the chain rule to get the exact differentiation. For solving this question, we have to remember that, if a line has a slope m, then the equation of tangent line will be y=mx.

Complete step by step answer:
Let us solve the question.
We have to find out the tangent line of the curve $y=\ln \left( 3x-5 \right)$.
As we know that whenever we have to find the tangent line or slope of a curve or a graph, first we have to differentiate that curve or graph with respect to x.
So let us differentiate the equation with respect to x.
The equation is $y=\ln \left( 3x-5 \right)$.
$\dfrac{dy}{dx}=\dfrac{d\left( \ln \left( 3x-5 \right) \right)}{dx}$
As we know that the differentiation of $\ln x$ is $\dfrac{1}{x}$.
Therefore, the differentiation of $\ln \left( 3x-5 \right)$ will be
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( \ln \left( 3x-5 \right) \right)}{dx}=\dfrac{1}{3x-5}\dfrac{d\left( 3x-5 \right)}{dx}$
According to chain rule, we will have to differentiate 3x-5.
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{3x-5}\dfrac{d\left( 3x-5 \right)}{dx}$
The differentiation of 3x-5 will be 3-0=3
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{3x-5}\times 3$
$\Rightarrow \dfrac{dy}{dx}=\dfrac{3}{3x-5}$
Now, we will put the value of x as 3 in $\dfrac{dy}{dx}=\dfrac{3}{3x-5}$ to get the exact value of tangent or slope at x=3.
$\Rightarrow {{\left. \dfrac{dy}{dx} \right|}_{x=3}}={{\left. \dfrac{3}{3x-5} \right|}_{x=3}}=\dfrac{3}{3\times 3-5}=\dfrac{3}{4}$
Hence, we get that the tangent is $\dfrac{3}{4}$ at x=3.
We know that if a line has a slope of m (say), then the equation of line will be y=mx.
Therefore, equation of tangent line whose tangent $m=\dfrac{3}{4}$ is
$y=mx=\dfrac{3}{4}x$
That is
$\Rightarrow 4y=3x$

Note: For solving this type of question, we should have a proper knowledge in calculus for finding the tangent line of any type of curve. Also, we should have a little bit of knowledge in differentiation topics.
In this type of problem, don’t forget to use the chain rule after the differentiation, otherwise the solution will be wrong.