Studdy AI Logo
Math  /  Trigonometry

QuestionMatch each trigonometric function with its derivative by entering the correct letter in the gi A. y=tan⁡(7x)y=\tan (7 x) B. y=tan⁡(x7)y=\tan \left(x^{7}\right) dydx=sec⁡2(x7)\frac{d y}{d x}=\sec ^{2}\left(x^{7}\right) C. y=cot⁡7(x)y=\cot ^{7}(x) dydx=−7cot⁡6(x)csc⁡(x)\frac{d y}{d x}=-7 \cot ^{6}(x) \csc (x) dydx=7x6sec⁡2(x7)\frac{d y}{d x}=7 x^{6} \sec ^{2}\left(x^{7}\right) D. y=7cot⁡(x)y=7 \cot (x) dydx=−7cot⁡6(x)csc⁡2(x)\frac{d y}{d x}=-7 \cot ^{6}(x) \csc ^{2}(x) E. Not the derivative of A,B,CA, B, C, or DD. dydx=−7csc⁡2(x)dydx=7sec⁡2(7x)\begin{array}{l} \frac{d y}{d x}=-7 \csc ^{2}(x) \\ \frac{d y}{d x}=7 \sec ^{2}(7 x) \end{array}

Studdy Solution

STEP 1

1. We are given a list of trigonometric functions and their derivatives.
2. We need to match each function with its correct derivative.
3. We will use known derivative rules for trigonometric functions and chain rule where applicable.

STEP 2

1. Calculate the derivative of each function.
2. Match each calculated derivative with the given options.
3. Identify any derivatives that do not match any function.

STEP 3

Calculate the derivative of each function:
A. y=tan⁡(7x) y = \tan(7x)
Using the chain rule, the derivative is: dydx=sec⁡2(7x)⋅7=7sec⁡2(7x) \frac{dy}{dx} = \sec^2(7x) \cdot 7 = 7 \sec^2(7x)
B. y=tan⁡(x7) y = \tan(x^7)
Using the chain rule, the derivative is: dydx=sec⁡2(x7)⋅7x6=7x6sec⁡2(x7) \frac{dy}{dx} = \sec^2(x^7) \cdot 7x^6 = 7x^6 \sec^2(x^7)
C. y=cot⁡7(x) y = \cot^7(x)
Using the chain rule, the derivative is: dydx=7cot⁡6(x)⋅(−csc⁡2(x))=−7cot⁡6(x)csc⁡2(x) \frac{dy}{dx} = 7 \cot^6(x) \cdot (-\csc^2(x)) = -7 \cot^6(x) \csc^2(x)
D. y=7cot⁡(x) y = 7 \cot(x)
The derivative is: dydx=7⋅(−csc⁡2(x))=−7csc⁡2(x) \frac{dy}{dx} = 7 \cdot (-\csc^2(x)) = -7 \csc^2(x)

STEP 4

Match each calculated derivative with the given options:
- For A: 7sec⁡2(7x) 7 \sec^2(7x) matches with dydx=7sec⁡2(7x) \frac{dy}{dx} = 7 \sec^2(7x) . - For B: 7x6sec⁡2(x7) 7x^6 \sec^2(x^7) matches with dydx=7x6sec⁡2(x7) \frac{dy}{dx} = 7x^6 \sec^2(x^7) . - For C: −7cot⁡6(x)csc⁡2(x) -7 \cot^6(x) \csc^2(x) matches with dydx=−7cot⁡6(x)csc⁡2(x) \frac{dy}{dx} = -7 \cot^6(x) \csc^2(x) . - For D: −7csc⁡2(x) -7 \csc^2(x) matches with dydx=−7csc⁡2(x) \frac{dy}{dx} = -7 \csc^2(x) .

STEP 5

Identify any derivatives that do not match any function:
- The derivative dydx=sec⁡2(x7) \frac{dy}{dx} = \sec^2(x^7) does not match any of the given functions A, B, C, or D.
The matches are: - A matches with dydx=7sec⁡2(7x) \frac{dy}{dx} = 7 \sec^2(7x) . - B matches with dydx=7x6sec⁡2(x7) \frac{dy}{dx} = 7x^6 \sec^2(x^7) . - C matches with dydx=−7cot⁡6(x)csc⁡2(x) \frac{dy}{dx} = -7 \cot^6(x) \csc^2(x) . - D matches with dydx=−7csc⁡2(x) \frac{dy}{dx} = -7 \csc^2(x) . - E corresponds to dydx=sec⁡2(x7) \frac{dy}{dx} = \sec^2(x^7) .

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord