Find (d)/(dv)(v^(4)-4sin v) Answer:

Apply Power Rule: Differentiate the function term by term. The function we are differentiating is v^(4)-4sin(v). We will apply the power rule to the first term and the derivative of the sine function to the second term.Power Rule for v^(4): Apply the power rule to the first term v^(4). The power rule states that (d/dv)(v^n) = nv^(n-1). Therefore, the derivative of v^(4) with respect to v is: (d/dv)(v^(4)) = 4v^(4-1) = 4v^3.Differentiate -4sin(v): Differentiate the second term -4sin(v). The derivative of sin(v) with respect to v is cos(v). Therefore, the derivative of -4sin(v) with respect to v is: (d/dv)(-4sin(v)) = -4cos(v).Combine Derivatives: Combine the derivatives of the individual terms. The derivative of the entire function v^(4)-4sin(v) with respect to v is the sum of the derivatives of its terms: (d/dv)(v^(4)-4sin(v)) = 4v^3 - 4cos(v).

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Find derivatives of using multiple formulae

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Q. Find

\frac{d}{d v}\left(v^{4}-4 \sin v\right)

\newline

Answer:

Apply Power Rule: Differentiate the function term by term. $\newline$ The function we are differentiating is $v^{4}-4\sin(v)$ . We will apply the power rule to the first term and the derivative of the sine function to the second term.
Power Rule for $v^{4}$ : Apply the power rule to the first term $v^{4}$ . The power rule states that $(\frac{d}{dv})(v^n) = nv^{n-1}$ . Therefore, the derivative of $v^{4}$ with respect to $v$ is: $(\frac{d}{dv})(v^{$4$}) = $4$v^{$4$$-1$} = $4$v^$3$.
Differentiate $-4\sin(v)$: Differentiate the second term $-4\sin(v)$. The derivative of $\sin(v)$ with respect to $v$ is $\cos(v)$. Therefore, the derivative of $-4\sin(v)$ with respect to $v$ is: $(\frac{d}{dv})(-4\sin(v)) = -4\cos(v)$.
Combine Derivatives: Combine the derivatives of the individual terms.$\newline$The derivative of the entire function $v^{4}-4\sin(v)$ with respect to $v$ is the sum of the derivatives of its terms:$\newline$$\frac{d}{dv}(v^{4}-4\sin(v)) = 4v^3 - 4\cos(v)$.