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Derivative of lnx?

TopTrigonometry is a branch of mathematics, which is used to show the relationship between the angles and sides of a triangle. In Trigonometry, we study the different types of trigonometric functions; here we will concentrate on the different type of Derivatives.
Some of the derivatives of trigonometric function are shown below:
= d / dy sin (y) = cos (y);
= d / dy cos (y) = - sin (y);
= d / dy cot (y) = - csc2 (y);
= d / dy ln (x) = 1 / x;
These all are different types of Derivatives of Trigonometric Functions.
Now we will see the derivative of lnx.
As we know, the derivative of lnx is 1 / x;
Let’s see prove of derivative of lnx.
First we write the ln x in the derivative form:
Proof = d / dx ln x = 1 /x;
We have to solve this derivative function by limit method.
In the limit form we can write it as:
= d / dx ln (x) = lim (a > 0) [ln (x + a) – ln (x)] / a = lim ln ((x + a) / x) / a;
In simple way we can write it as:
= lim (1 / a) ln (1 + a / x) = lim [ln (1 + a / x) (1 / a)];
Now let u = a / x and substitute the values then we get:
= lim (a > 0) [ln (1 + a)(1 / ax)] = 1 / x ln [lim(a > 0) (1 + a)(1 / a)];
If we further solve then we get:
= 1 / x ln (e); by the definition of e;
So we can write it as:
= 1 / x.
This is how to prove the derivative of lnx.