Find the nth derivative of 2*sin(3*t-1)

asked by guest
on Jan 17, 2025 at 12:34 pm



MathBot Answer:

The first few derivatives of y=2sin(3t1)y = 2 \sin\left( 3 t - 1 \right) with respect to tt are: dydt=6cos(3t1)d2ydt2=18sin(3t1)d3ydt3=54cos(3t1)d4ydt4=162sin(3t1)d5ydt5=486cos(3t1) \begin{aligned} \frac{d y}{d t} &= 6 \cos{\left(3 t - 1 \right)}\\ \frac{d^{2} y}{d t^{2}} &= - 18 \sin{\left(3 t - 1 \right)}\\ \frac{d^{3} y}{d t^{3}} &= - 54 \cos{\left(3 t - 1 \right)}\\ \frac{d^{4} y}{d t^{4}} &= 162 \sin{\left(3 t - 1 \right)}\\ \frac{d^{5} y}{d t^{5}} &= 486 \cos{\left(3 t - 1 \right)} \end{aligned}

Given a positive integer nn, let rr denote the remainder when nn is divided by 44. The nth n^{\text{th}} derivative of 2sin(3t1)2 \sin{\left(3 t - 1 \right)} with respect to tt is given by the formula: dndtn[2sin(3t1)]={23nsin(3t1) when r=0(1)23ncos(3t1) when r=123nsin(3t1) when r=2(1)23ncos(3t1) when r=3 \frac{d^n}{d t^n} \left[ 2 \sin{\left(3 t - 1 \right)} \right] = \begin{cases} 2 \cdot 3^{n} \sin{\left(3 t - 1 \right)} &\text{ when } r = 0\\ \left(-1\right) 2 \cdot 3^{n} \cos{\left(3 t - 1 \right)} &\text{ when } r = 1\\ 2 \cdot 3^{n} \sin{\left(3 t - 1 \right)} &\text{ when } r = 2\\ \left(-1\right) 2 \cdot 3^{n} \cos{\left(3 t - 1 \right)} &\text{ when } r = 3 \end{cases}


In general, the nnthn^{\text{nth}} derivative of asin(bx+c)a \sin( b x + c) is given by the formula: dndxn[asin(bx+c)]={abnsin(bx+c) when r=0abncos(bx+c) when r=1abnsin(bx+c) when r=2abncos(bx+c) when r=3 \frac{d^n}{d x^n} \left[ a \sin(b x + c) \right] = \begin{cases} a b^n \sin(b x + c) &\text{ when } r = 0\\ a b^n \cos(b x + c) &\text{ when } r = 1\\ -a b^n \sin(b x + c) &\text{ when } r = 2\\ -a b^n \cos(b x + c) &\text{ when } r = 3 \end{cases} In the case of 2sin(3t1)2 \sin{\left(3 t - 1 \right)}, a=2a = 2, b=3b = 3 and c=1c = -1.