Explained: "Evaluate the derivatives of the following functions. f(x)=sin⁡(tan−1(ln⁡x))" - Plainmath

Brennan Flores

Answered question

2021-09-23

Evaluate the derivatives of the following functions.

f (x) = \sin (\tan^{- 1} (\ln x))

Answer & Explanation

irwchh

Skilled2021-09-24Added 102 answers

Step 1
Given:

f (x) = \sin (\tan^{- 1} (\ln x))

We know that

\frac{d}{d x} (\sin x) = \cos x

...(1)

\frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}

...(2)

\frac{d}{d x} (\ln x) = \frac{1}{x}

...(3)
Step 2
So, by using equation(1), (2) and (3)

f^{'} (x) = \frac{d}{d x} (\sin (\tan^{- 1} (\ln x)))

= \cos (\tan^{- 1} (\ln x)) \frac{d}{d x} (\tan^{- 1} (\ln x))

= \cos (\tan^{- 1} (\ln x)) \frac{1}{1 + {(\ln x)}^{2}} \frac{d}{d x} (\ln x)

= \cos (\tan^{- 1} (\ln x)) \frac{1}{1 + \ln^{2} x} (\frac{1}{x})

= \frac{\cos (\tan^{- 1} (\ln x))}{x (1 + \ln^{2} x)}

Hence, derivative of given function can be find as above.

Jeffrey Jordon

Expert2022-03-13Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?
How do I find the y-intercept of a parabola?
What are the vertices of $9 x^{2} + 16 y^{2} = 144$ ?
How to determine the rate of change of a function?
Why are the tangents for 90 and 270 degrees undefined?
How to find the center and radius of the circle $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
What is multiplicative inverse of a number?
How to find the continuity of a function on a closed interval?
How do I find the tangent line of a function?
How to find vertical asymptotes using limits?
How to find the center and radius of the circle $x^{2} - 12 x + y^{2} + 4 y + 15 = 0$ ?
Let f be a function so that (below). Which must be true?
I. f is continuous at x=2
II. f is differentiable at x=2
III. The derivative of f is continuous at x=2
(A) I (B) II (C) I and II (D) I and III (E) II and III
How to find the center and radius of the circle given $x^{2} + y^{2} + 8 x - 6 y = 0$ ?
How to find the center and radius of the circle $x^{2} + y^{2} + 4 x - 8 y + 4 = 0$ ?
How to identify the center and radius of the circle ${(x + 3)}^{2} + {(y - 8)}^{2} = 16$ ?

Didn't find what you were looking for?