Find the derivative of the function.r=cscx−3sinx+tanxr=\csc x-3\sin x+\tan xr=cscx−3sinx+tanx

− csc ⁡ x cot ⁡ x − 3 cos ⁡ x + sec ⁡ 2 x -\csc x\cot x-3\cos x+\sec^2x − csc x cot x − 3 cos x + sec 2 x

Find the derivative of the function. r = csc x − 3 sin x + tan x r=\csc x-3\sin x+\tan x r = csc x − 3 sin x + tan x

this problem, we're asked to find the derivative of the function r equals cosecant x minus three sine x plus tangent of x. Now here, since this is kind of the first time that we're working with the derivatives of all of these trig functions, as well as all of the rules for derivatives that we already know. I'm going to go ahead and fully break this down for you so that you can see clearly all of the rules that I'm using. So let's go ahead and jump in here. So r prime, our derivative that we want to find here, is going to be equal to the derivative d d x of cosecant x minus, because we're subtracting here, so this is our difference rule for derivatives. We are subtracting. And since I have a constant, I can go ahead and pull that constant out to the front. That is our constant multiple rule times the derivative d d x of sine x. Then I'm adding that together using my sum rule for derivatives, d d x of tangent x, that last term. So we've combined quite a few different rules here. Even if it's not immediately clear when looking at this, we are still using our sum and difference rules the same way that we have regardless of if we had trig functions there or not, as well as our constant multiple rule. Now we just need to apply our rules for derivatives of trig functions, which is what we just learned. So let's go ahead and take these derivatives. Now the derivative of the cosecant, remember that the derivative of all of our co functions is always going to be negative. So I already know I need a negative there. And the derivative of cosecant specifically is going to be equal to the cotangent of x times the cosecant of x. So we do have that original function in here, but it's being multiplied by that cotangent. Then here I'm subtracting three times the derivative of sine x, which we know to be cosine of x. And then finally, I'm adding the derivative of the tangent of x, which we know is going to be equal to the secant squared of x. Now we're all done here. This is our full derivative. And this is our answer, negative cotangent x cosecant x minus three cosine of x plus the secant squared of x. Remember that all of the rules for derivatives that we've already learned in the past still apply even when working with all of these trig functions as well. Feel free to let me know if you have any questions, and I will see you in the next video.

Find the derivative of the function. r = csc ⁡ x − 3 sin ⁡ x + tan ⁡ x r=\csc x-3\sin x+\tan x r = csc x − 3 sin x + tan x

− csc ⁡ x cot ⁡ x − 3 cos ⁡ x + sec ⁡ 2 x -\csc x\cot x-3\cos x+\sec^2x − csc x cot x − 3 cos x + sec 2 x

csc ⁡ x cot ⁡ x + cos ⁡ x + sec ⁡ 2 x \csc x\cot x+\cos x+\sec^2x csc x cot x + cos x + sec 2 x

− csc ⁡ x cot ⁡ x − cos ⁡ x + sec ⁡ 2 x -\csc x\cot x-\cos x+\sec^2x − csc x cot x − cos x + sec 2 x

− cot ⁡ x − 3 cos ⁡ x + sec ⁡ 2 x -\cot x-3\cos x+\sec^2x − cot x − 3 cos x + sec 2 x

12. Trigonometric Functions

Derivatives of Trig Functions

Business Calculus

12. Trigonometric Functions

Derivatives of Trig Functions

Struggling with Business Calculus?

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Multiple Choice

Find the derivative of the function.
$r=\csc x-3\sin x+\tan x$

$\csc x\cot x+\cos x+\sec^2x$

$-\csc x\cot x-\cos x+\sec^2x$

$-\csc x\cot x-3\cos x+\sec^2x$

$-\cot x-3\cos x+\sec^2x$

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is r = csc(x) - 3sin(x) + tan(x).

Step 2: Recall the derivatives of the trigonometric functions involved: (1) The derivative of csc(x) is -csc(x)cot(x), (2) The derivative of sin(x) is cos(x), and (3) The derivative of tan(x) is sec^2(x).

Step 3: Apply the derivative to each term of the function r. For the first term, differentiate csc(x) to get -csc(x)cot(x). For the second term, differentiate -3sin(x) to get -3cos(x). For the third term, differentiate tan(x) to get sec^2(x).

Step 4: Combine the results from Step 3. The derivative of r is the sum of the derivatives of each term: -csc(x)cot(x) - 3cos(x) + sec^2(x).

Step 5: Simplify the expression if necessary. In this case, the derivative is already in its simplest form: -csc(x)cot(x) - 3cos(x) + sec^2(x).

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