Find (d)/(dv)(5v^(3)+5sin v) Answer:

Identify Function: Identify the function to differentiate. We are given the function f(v) = 5v^(3) + 5sin(v), and we need to find its derivative with respect to v.Apply Sum Rule: Apply the sum rule for differentiation. The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, we can differentiate each term separately.Differentiate First Term: Differentiate the first term 5v^(3). Using the power rule, which states that the derivative of v^n is n*v^(n-1), we differentiate 5v^(3) to get 3*5v^(3-1) = 15v^(2).Differentiate Second Term: Differentiate the second term 5sin(v). The derivative of sin(v) with respect to v is cos(v), so the derivative of 5sin(v) is 5cos(v).Combine Derivatives: Combine the derivatives of both terms. The derivative of the function f(v) = 5v^(3) + 5sin(v) is the sum of the derivatives of its terms, which we found in steps 3 and 4. Therefore, the derivative is 15v^(2) + 5cos(v).

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Find $\frac{d}{d v}\left(5 v^{3}+5 \sin v\right)$ $\newline$ Answer:

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Algebra 1

Multiplication with rational exponents

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Q. Find

\frac{d}{d v}\left(5 v^{3}+5 \sin v\right)

\newline

Answer:

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $f(v) = 5v^{3} + 5\sin(v)$ , and we need to find its derivative with respect to $v$ .
Apply Sum Rule: Apply the sum rule for differentiation. $\newline$ The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, we can differentiate each term separately.
Differentiate First Term: Differentiate the first term $5v^{3}$ . Using the power rule, which states that the derivative of $v^n$ is $n*v^{(n-1)}$ , we differentiate $5v^{3}$ to get $3*5v^{(3-1)} = 15v^{2}$ .
Differentiate Second Term: Differentiate the second term $5\sin(v)$ . The derivative of $\sin(v)$ with respect to $v$ is $\cos(v)$ , so the derivative of $5\sin(v)$ is $5\cos(v)$ .
Combine Derivatives: Combine the derivatives of both terms. $\newline$ The derivative of the function $f(v) = 5v^{3} + 5\sin(v)$ is the sum of the derivatives of its terms, which we found in steps $3$ and $4$ . Therefore, the derivative is $15v^{2} + 5\cos(v)$ .