Equal derivatives Verify that the functions f(x) = tan² x ...

Question

Equal derivatives Verify that the functions f(x) = tan² x and g(x) = sec² x have the same derivative. What can you say about the difference f - g? Explain.

Accepted Answer

Hello, in this video, we want to suppose that the function M X is equal to cotangent squared, and the function n of X is equal to cos 2. If the function M and function N have the same derivative, what does this imply about the expression MX minus NX? Now, in order to approach this problem, the first thing we want to do is you want to verify the first part of the statement. We want to verify that MX and NX have the same derivative. So, the function M X is equal to cotangent squared of x. Now, we want to go ahead and take the derivative of this function. Now, recall that by definition, the derivative of cotangent is equal to negative cosequent squared. Of X What this means is that if we take the derivative of cotangent, we will get negative cosequin squared. But keep in mind that we have this exponent of 2, so in order to take the derivative of M of X, we will go ahead and use the chain rule. So, we will take the derivative of the outermost function, which is going to be the exponent of 2. That is going to give us 2 multiplied by cotangent of X. Then we will multiply this product. We will multiply this by the derivative of cotangent of X, and the derivative of cotangent of X will be negative cosequent squared of X. By multiplying the coefficients together, we can further rewrite this to be negative to. Cotangent of X. Multiplied by coin squad of X, and this is going to be the derivative M X. Now, what we want to do is you want to take the derivative of the function N of X. Now, N X is equal to cosin2 X. Let's go ahead and recall the derivative of cossekin squared. cosequant, the derivative of cosequant of X is equal to negative cosequant. Cotangent of X. And again, since we have this exponent of 2, we will go ahead and take the derivative by using the chain rule. That is going to be 2 multiplied by cosequent of X, multiplied by the derivative of cosequent, which will be negative cosequant of X, cotangent of X. By multiplying the coefficients together, and by combining the cosequent terms, we can rewrite this to be negative2 multiplied by cotangent of X, multiplied by cosequent squared of X, and this is going to be the derivative N of X. By comparing both of the derivatives together, we can verify that both of the derivatives are equal to each other. So, what does this imply about the original statement M X minus N of X? Well, the first thing we need to do is we need to figure out what kind of output this function will give us. So by plugging in the given functions M and N, we can rewrite the statement to be cotangent squared X minus cosequent squared X. Now, by the Pythagorean identity, we can rewrite cosin squared X to be 1 plus cotangent squared X. So, if we use this Pythagorean identity, we can rewrite the statement to be cotangent squared X minus 1 plus cotangent squared X. By distributing a -1, we can rewrite the statement to be cotangent squared of X. 1 minus cotangent square of X, and that is going to give us an output of -1. So what this means is that the function M X minus N of X is going to be the constant function -1. And if we were to take the derivative of this function, that is going to give us a zero derivative, meaning that the original function is a constant function. With that being said, the solution to this problem is going to be D. So, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Equal derivatives Verify that the functions f(x) = tan² x and g(x) = sec² x have the same derivative. What can you say about the difference f - g? Explain.

Key Concepts

Derivatives

Trigonometric Functions

Difference of Functions

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