(d)/(dx)[sin^(7)(x)]=? Choose 1 answer: (A) 7sin^(6)(x)cos(x) (B) 7cos^(6)(x) (C) 7x^(6)cos(x^(7)) (D) cos(7x^(6))

Apply Chain Rule: Apply the chain rule to the function sin^(7)(x). The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x))g'(x). In this case, f(u) = u^7 and g(x) = sin(x), so we need to find the derivatives f'(u) and g'(x).Derivative of u^7: Find the derivative of f(u) = u^7 with respect to u. Using the power rule, (d/du)(u^n) = nu^(n-1), we get: (d/du)(u^7) = 7u^(7-1) = 7u^6.Derivative of sin(x): Find the derivative of g(x) = sin(x) with respect to x. The derivative of sin(x) with respect to x is cos(x). (d/dx)(sin(x)) = cos(x).Apply Chain Rule Again: Apply the chain rule using the derivatives from Step 2 and Step 3. The derivative of sin^(7)(x) with respect to x is: (d/dx)(sin^(7)(x)) = 7(sin(x))^6 * cos(x).Match Result: Match the result with the given options. The correct answer is (A) 7sin^(6)(x)cos(x), which matches our result from Step 4.

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(d)/(dx)[sin^(7)(x)]=?
Choose 1 answer:
(A)
7sin^(6)(x)cos(x)
(B)
7cos^(6)(x)
(C)
7x^(6)cos(x^(7))
(D)
cos(7x^(6))

$\frac{d}{d x}\left[\sin ^{7}(x)\right]=?$ $\newline$ Choose $1$ answer: $\newline$ (A) $7 \sin ^{6}(x) \cos (x)$ $\newline$ (B) $7 \cos ^{6}(x)$ $\newline$ (C) $7 x^{6} \cos \left(x^{7}\right)$ $\newline$ (D) $\cos \left(7 x^{6}\right)$

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Find derivatives of using multiple formulae

Full solution

\frac{d}{d x}\left[\sin ^{7}(x)\right]=?

\newline

Choose

1

answer:

\newline

(A)

7 \sin ^{6}(x) \cos (x)

\newline

(B)

7 \cos ^{6}(x)

\newline

(C)

7 x^{6} \cos \left(x^{7}\right)

\newline

(D)

\cos \left(7 x^{6}\right)

Apply Chain Rule: Apply the chain rule to the function $\sin^{7}(x)$ . The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$ . In this case, $f(u) = u^{7}$ and $g(x) = \sin(x)$ , so we need to find the derivatives $f'(u)$ and $g'(x)$ .
Derivative of $u^7$ : Find the derivative of $f(u) = u^7$ with respect to $u$ . Using the power rule, $(\frac{d}{du})(u^n) = nu^{(n-1)}$ , we get: $(\frac{d}{du})(u^7) = 7u^{(7-1)} = 7u^6$ .
Derivative of sin(x): Find the derivative of $g(x) = \sin(x)$ with respect to $x$ . $\newline$ The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$ . $\newline$ $\frac{d}{dx}(\sin(x)) = \cos(x)$ .
Apply Chain Rule Again: Apply the chain rule using the derivatives from Step $2$ and Step $3$ . $\newline$ The derivative of $\sin^{7}(x)$ with respect to $x$ is: $\newline$ $\frac{d}{dx}(\sin^{7}(x)) = 7(\sin(x))^6 \cdot \cos(x)$ .
Match Result: Match the result with the given options. $\newline$ The correct answer is $(A) 7\sin^{6}(x)\cos(x)$ , which matches our result from Step $4$ .