Verifying derivative formulas Verify the following derivative ...

Question

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

d/dx (csc x) = -csc x cot x

Accepted Answer

Welcome back once another video. Determine the derivative of the function F of X equals tangent of X using the quotient rule. We're given 4 answer choices A says sequent X tangent X, B, 1 divided by 1 + X2, C, negative cos 2 X, and D 22 X. So what we're going to do is simply look at the given function. It says that F of X is equal to tangent of X, and we want to evaluate as derivative using the quotient rule. So let's recall the definition of tangent. It is basically the ratio between sine and cosine. So we're going to rewrite tangent as a fractions of x divided by cosine of X. And then we want to differentiate to the given function and we have to recall the quotient rule. According to the quotient rule, first of all, we take the derivative of the top functions. We're going to take the derivative of sine of X, and we're going to multiply that by the bottom function, which is cosine of X. Then we are subtracting the top function which is sign of X multiplied by the derivative of the bottom function, so the derivative of cosine of X, and all of that difference must be divided by the square of the bottom function, so cosine squared of x. And from here we want to recall that according to the table of basic derivatives, the derivative of sine is equal to cosine, and the derivative of cosine is equal to negative sign. So we're going to use those in our expression. We end up with cosine multiplied by cosine, which is cosine squared of x, and we are subtracting sign multiplied by negative sign. So if we're subtracting a negative term, this becomes a positive sign, and we get sine squared of x. And now at the bottom of the fraction we still have cosine squared of X. Now we can notice that in the numerator, according to the Pythagorean identity, we end up with one sine square plus cosine square. This gives us one, and then in the denominator, we still have cosine square. X. And now if we rewrite it as 1 divided by cosine of x and write the square outside, we can notice that the term inside is sequent, right, so that'd be sequent squared of x. Which is consistent with the definition of the derivative of tangent based on the table of basic derivatives, and our final answer corresponds to the answer choice D 2 square of X. Thank you for watching.

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (csc x) = -csc x cot x

Key Concepts

Quotient Rule

Cosecant Function

Trigonometric Identities

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