Verifying derivative formulas Verify the following derivative ...

Question

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (sec x) = sec x tan x

Accepted Answer

Welcome back, everyone. In this problem, we want to apply the quotient rule to find the derivative of the function F X equals the cotangent of X. A says it's negative coex 2 X, B 62 X D negative co X X where cut means the cotangent, and D-1 divided by 1 + X2. Now, what does it mean for us to apply the quotient rule to find the derivative of the function? Well, first of all, note that the cotangent, OK, core tangent function is really the reciprocal of the tangent function. So we could rewrite cut X as one divided by the tangent of X, but what do we know about the tangent of X? Remember, the tangent is the ratio of the sine to the cosine function. So that means the reciprocal of that will be equal to the cosine function plus X divided by the sine of X. Now we've been able to express F of X as a caution and thus we'll be able to apply the cauttient rule. So what do we know about the quotient rule? Well, recall, it basically tells us that if we have a function Y that's equal to U divided by V, where U and V are differentiable functions, then the derivative of Y is equal to U U V. Apologies. Minusuv divided by v squad where U and V are the derivatives of U and V respectively. Now for our quotientu represents the numerator, V represents the denominator. So if we can differentiate both of those, then we should be able to plug them into the quotient rule and find the derivative of Y. Now what do we know? Well, we know that U equals the cosine of X. And we know that V equals the sign of X. The derivative of U will be equal to negative sine X. That's the derivative of the cosine, and the derivative of V will be the cosine of X. That's the derivative of sine. Know that we have UV and its derivatives, we can apply the quotient rule. Because no, that means then that the derivative of Y equals UV. So that's negative sin X multiplied by sin X. Minus UV, so that's the cosine of X multiplied by the cosine of X, OK. And all of this is being divided by V, the sin of X squared or sin squared X. Now, if we expand our terms here and factor out to a negative sign, we're gonna have sin X multiplied by sin X, which is sin squared X, and co X multiplied by cos X, which is cos squared X, and we can write that the both of them are being added since we factored or on negative sign. Now could we simplify this expression any further? Well, not in a numerator, we have the trigonometric identity. The fundamental trigonometric identity, which says that the square of the sine function plus the square of the cosine function for the same argument is equal to one. So with that idea, then we could rewrite our numerator as 1, so that's -1 divided by squared X. Now, recall, OK, recall that. The reciprocal, yeah. The the reciprocal sign function is the cocant, OK? In other words, the cos of X is equal to 1 divided by the sine of X. So with that in mind then, we could take a step further and write the derivative of Y, OK, or in this case, our function as F of X or the derivative of F of X to be equal to the negative value of the square of the core C can of X. Therefore, this is the derivative of the function F of X equals the quote tent of X. If we look back at our answer choices, that means A is the correct answer. Thanks for watching everyone. I hope this video helped.

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (sec x) = sec x tan x

Key Concepts

Quotient Rule

Trigonometric Derivatives

Chain Rule

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