Evaluate the derivative of the following functions.
f(x)...

Question

Evaluate the derivative of the following functions.

f(x) = sin(tan^-1 (ln x))

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the derivative of F of X is equal to the cosine of tangent to the power of -1 of 10 multiplied by the natural log of X. Awesome. So, it's important to note that tangent to the power of -1 means inverse tangent. So we're dealing with a pretty complex looking function here, so it has a lot of moving parts, which we'll discuss in a bit as we solve, but we need to note that for this particular problem, we're ultimately trying to solve for the first derivative. We're trying to find the derivative of F of X. So we're trying to find F of X. So with that said, Our first step, first off, let us let us state that U is equal to inverse tangent, so tangent to the power of -1. Of 10 multiplied by the natural log of X. So let us say that U equals that. That U equals inverse tangent of 10 multiplied by the natural log of X, and let us also state that F of X is going to be equal to cosine of u. Fantastic. So now, our next step is we need to find the derivative of this cosine function. So we need to take D by DX, so we're taking the derivative with respect to X. So D by DX of cosine of U, which will be equal to negative sine of u multiplied by Du by DX, fantastic. So now, We need to differentiate F of X, so we need to state that F of X, where this prime symbol means it's the first derivative of this function of F of X, kind of looks like a, so it kind of looks like an apostrophe in the power. So that's what the prime symbol is. So F of X is gonna be equal to negative sign of. Inverse tangent of 10 multiplied by the natural log of X, multiplied by D by DX. Once again we're taking the derivative with respect to X of inverse tangent of 10 multiplied by the natural log of X. Fantastic. So now let us consider D by DX of the inverse tangent of you. Awesome. So now we're focusing on DX D by DX of inverse tangent of you. Fantastic. Or rather, let's say, so like, let us consider, so let's take a step back here. So let us consider D by DX of inverse tangent of 10 multiplied by the natural log of X. So let us know that the derivative of an inverse tangent function and the natural logamithic function will be that D by DX of, and this is where we can write that inverse tangent of U. So D D by DX of inverse tangent of you will be equal to 1 divided by 1 plus u squared multiplied by DU by DX. So, therefore, we could go ahead and write that D by DX of the natural log of U is equal to 1 divided by U, multiplied by DU by DX. So now, using this information, we can recall and use the chain rule, so we can apply the chain rule to help us perform this derivative, so we could take D by DX of inverse tangent of 10 multiplied by the natural log of X, which is equal to 1 divided by 1 plus. 10 multiplied by the natural log of X2. Multiplied by D by DX of 10 multiplied by the natural log of X, fantastic. And this will be equal to Drum roll, one divided by one plus. 10 multiplied by the natural log of X 2 multiplied by 10, multiplied by 1 divided by X, which will be all equal to 10 divided by X. Multiplied by, so be 10, all divided by X multiplied by 1 plus 10 multiplied by the natural log of X squared. Fantastic. Awesome, we are on a roll here. So now, Our next step that we need to consider is we need to take and focus our attention on The following meaning to state that F of X is equal to negative, sign of inverse tangent of 10 multiplied by the natural log of X, multiplied by D by DX of inverse tangent. Of 10 multiplied by the natural log of X. So we need to consider this expression, which we obtained earlier, and we need to substitute in D by DX of inverse tangent of 10. Multiplied by the natural log of X, which is equal to once again 10 divided by X multiplied by 1 plus 10 multiplied by the natural log of X to the power of 2. Fantastic. So now, with that in mind, We can go ahead and write that F of X is equal to negatives of. Inverse tangent of 10 multiplied by the natural log of X. Multiplied by D by DX of. Inverse tangent of 10 multiplied by the natural log of X. Fantastic. So now, Our next step is we need to take F of X is equal to negatives. Of inverse tangent of 10 multiplied by the natural log of X. Awesome, and we need to take this, and we need to multiply it by 10 divided by X multiplied by 1 plus 10 multiplied by the natural log of X squared. And when we do that, We will find that when we simplify that F of X is equal to negative. 10 multiplied by sign of. Inverse tangent of 10 multiplied by the natural log of X. All divided by X multiplied by 1 + 10 multiplied by the natural log of X squared, and that is our final answer for the derivative of F of X. Hooray, we did it. So therefore, without a doubt, our final answer for the first derivative of this function of F of X will be F of X is equal to negative 10 multiplied by sine of inverse tangent of 10 multiplied by the natural log of X, all divided by X multiplied by 1 plus 10 multiplied by the natural log of x to the power of 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the derivative of the following functions.
f(x) = sin(tan^-1 (ln x))

Key Concepts

Chain Rule

Inverse Trigonometric Functions

Natural Logarithm

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