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

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

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Write a formula for an arithmetic sequence

Full solution

Q. Given that

y=u^{3}+3

, find

\frac{d}{d u}\left(3 y^{2}+4 \sin u\right)

in terms of only

u

.

\newline

Answer:

Find Derivative of $y$ : First, we need to find the derivative of $y$ with respect to $u$ , since $y$ is a function of $u$ . The given function is $y = u^3 + 3$ . $\newline$ Using the power rule for differentiation, the derivative of $u^3$ with respect to $u$ is $3u^2$ , and the derivative of a constant is $0$ . $\newline$ So, $y$ $0$ .
Derivative of $3y^2$ : Next, we need to find the derivative of $3y^2$ with respect to $y$ . Using the power rule again, the derivative of $y^2$ with respect to $y$ is $2y$ . So, $\frac{d(3y^2)}{dy} = 3 \times 2y = 6y$ .
Apply Chain Rule: Now, we apply the chain rule to find the derivative of $3y^2$ with respect to $u$ . The chain rule states that $\frac{d(f(g(u)))}{du} = f'(g(u)) \cdot g'(u)$ . Here, $f(y) = 3y^2$ and $g(u) = y$ , so we have $\frac{d(3y^2)}{du} = \frac{d(3y^2)}{dy} \cdot \frac{dy}{du} = 6y \cdot 3u^2$ .
Substitute $y$ into Expression: We substitute $y = u^3 + 3$ into the expression we found in the previous step to express it in terms of $u$ . So, $\frac{d(3y^2)}{du} = 6(u^3 + 3) \cdot 3u^2$ .
Simplify Expression: Now, we simplify the expression we found in the previous step. $\newline$ $\frac{d(3y^2)}{du} = 6(u^3 + 3) \times 3u^2 = 18u^2(u^3 + 3)$ .
Derivative of $4\sin u$ : Next, we find the derivative of $4\sin u$ with respect to $u$ . The derivative of $\sin u$ with respect to $u$ is $\cos u$ . So, $\frac{d(4\sin u)}{du} = 4\cos u$ .
Add Derivatives: Finally, we add the derivatives we found for $3y^2$ and $4\sin u$ to find the derivative of the entire expression $3y^2 + 4\sin u$ with respect to $u$ . $\frac{d}{du}(3y^2 + 4\sin u) = \frac{d(3y^2)}{du} + \frac{d(4\sin u)}{du} = 18u^2(u^3 + 3) + 4\cos u.$

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