Given that y=3u^(2)+4, find (d)/(du)(2u^(5)-4sin y) in terms of only u. Answer:

Identify Function: Identify the function that needs to be differentiated and the function y in terms of u.Differentiate Power Rule: Differentiate the function 2u^(5) with respect to u using the power rule, which states that d/du[u^n] = n*u^(n-1). Calculation: d/du[2u^(5)] = 2*5*u^(5-1) = 10u^(4)Apply Chain Rule: Recognize that y is a function of u, so when differentiating -4sin(y) with respect to u, we need to use the chain rule. The chain rule states that d/du[f(g(u))] = f'(g(u)) * g'(u).Differentiate y=3u^(2)+4: Differentiate y=3u^(2)+4 with respect to u to find dy/du. Calculation: dy/du = d/du[3u^(2)+4] = 6uApply Chain Rule: Apply the chain rule to differentiate -4sin(y) with respect to u. Calculation: d/du[-4sin(y)] = -4cos(y) * dy/du = -4cos(y) * 6u = -24u*cos(y)Substitute y=3u^(2)+4: Substitute y=3u^(2)+4 into the expression -24u*cos(y) to express it in terms of u. Calculation: -24u*cos(3u^(2)+4)Combine Derivatives: Combine the derivatives of 2u^(5) and -4sin(y) to get the final derivative in terms of u. Calculation: d/du[2u^(5)-4sin(y)] = 10u^(4) - 24u*cos(3u^(2)+4)

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Given that
y=3u^(2)+4, find
(d)/(du)(2u^(5)-4sin y) in terms of only
u.
Answer:

Given that $y=3 u^{2}+4$ , find $\frac{d}{d u}\left(2 u^{5}-4 \sin y\right)$ in terms of only $u$ . $\newline$ Answer:

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Q. Given that

y=3 u^{2}+4

, find

\frac{d}{d u}\left(2 u^{5}-4 \sin y\right)

in terms of only

u

\newline

Answer:

Identify Function: Identify the function that needs to be differentiated and the function $y$ in terms of $u$ .
Differentiate Power Rule: Differentiate the function $2u^{5}$ with respect to $u$ using the power rule, which states that $\frac{d}{du}[u^n] = n\cdot u^{(n-1)}$ . $\newline$ Calculation: $\frac{d}{du}[2u^{5}] = 2\cdot 5\cdot u^{(5-1)} = 10u^{4}$
Apply Chain Rule: Recognize that $y$ is a function of $u$ , so when differentiating $-4\sin(y)$ with respect to $u$ , we need to use the chain rule. The chain rule states that $\frac{d}{du}[f(g(u))] = f'(g(u)) \cdot g'(u)$ .
Differentiate $y=3u^{2}+4$ : Differentiate $y=3u^{2}+4$ with respect to $u$ to find $\frac{dy}{du}$ . $\newline$ Calculation: $\frac{dy}{du} = \frac{d}{du}[3u^{2}+4] = 6u$
Apply Chain Rule: Apply the chain rule to differentiate $-4\sin(y)$ with respect to $u$ . $\newline$ Calculation: $\frac{d}{du}[-4\sin(y)] = -4\cos(y) \cdot \frac{dy}{du} = -4\cos(y) \cdot 6u = -24u\cos(y)$
Substitute $y=3u^{2}+4$ : Substitute $y=3u^{2}+4$ into the expression $-24u\cos(y)$ to express it in terms of $u$ . $\newline$ Calculation: $-24u\cos(3u^{2}+4)$
Combine Derivatives: Combine the derivatives of $2u^{5}$ and $-4\sin(y)$ to get the final derivative in terms of $u$ . $\newline$ Calculation: $\frac{d}{du}[2u^{5}-4\sin(y)] = 10u^{4} - 24u\cos(3u^{2}+4)$