47–56. Derivatives of inverse functions at a point Consider th...

Question

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=tan x; (1,π/4)

Accepted Answer

Welcome back, everyone. For the function G of X equals cotangent of 21 X, determine the derivative of the inverse function at the 1. pi divided by 84 without calculating the inverse. We're given 4 answer choices A 1 divided by 21, B 1 divided by 42, C-1 divided by 21, and D-1 divided by 42. In the context of this problem we're given a point. Let's suppose that it has a form of a comb. Our A is 1, and our b coordinate is I divided by 84, and we want to recall the relationship between the derivative of the inverse function and the derivative of the parent function. So if we want to evaluate. The derivative of the inverts function. At a point A, right, because we want to evaluate it at X equals 1, we're going to say that j inverse derivative at point A. According to the identity, this is going to be equal to 1 divided by the derivative of the parent function at point B, and that's all that we have to do. So now let's substitute the values. We're going to get J in verse. Derivative at point A, which is 1, is equal to 1 divided by G derivative at point B, which is 5 divided by 84. So now we have to identify the derivative of J. Let's differentiate J of X, which corresponds to the derivative of cotangent of 21 X, right? So what is the derivative of cotangent? We have to recall its negative coc 2, right? And our angle is 21 X, meaning we have to apply the chain rules. So we're multiplying by the derivative of 21 X. Now the derivative of 21 X is 21 and we end up with negative 21 OC2 of 21 X. Our next step is to evaluate J at pi divided by 84. So now we take our J. And substitute pi divided by 844 X. We're going to get -21. Multiplied by cosy and squared. Of 21 multiplied by Pi divided by 84. This gives us negative 21 multiplied by cosic squared of. I divided by 4, we can simplify the angle, right? And a cosequence of pi divided by 4 is 1 divided by. Square root of 2. Actually it's square root of 2 itself, right? So to be accurate, let's write it down. Cos of pi divided by 4 is square root of 2, and now we are going to square it, right? So we get -21 multiplied by 2. So we end up with -42 and that's all that we have to do. So now if we get back to our original expression, we can say that J inverse derivative at 0.1 is simply equal to 1 divided by -42. So we can Factor out the negative sign in front of the fraction, and this gives us -1 divided by 42, meaning the correct answer choice to this problem corresponds to the answer choice D. Thank you for watching.

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=tan x; (1,π/4)

Key Concepts

Inverse Functions

Derivative of Inverse Functions

Trigonometric Functions and Their Derivatives

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