Find the 20th derivative of Y cos(2x). f (20)(x)
The Correct Answer and Explanation is:
To find the 20th derivative of the function y=cos(2x)y = \cos(2x)y=cos(2x), we can first explore the pattern in the derivatives of cosine functions, especially those with arguments involving multiples of xxx.
Step-by-Step Approach:
The function is:y=cos(2x)y = \cos(2x)y=cos(2x)
We begin by calculating the first few derivatives of y=cos(2x)y = \cos(2x)y=cos(2x).
- First derivative: ddx[cos(2x)]=−2sin(2x)\frac{d}{dx} [\cos(2x)] = -2\sin(2x)dxd[cos(2x)]=−2sin(2x)
- Second derivative: ddx[−2sin(2x)]=−2⋅2cos(2x)=−4cos(2x)\frac{d}{dx} [-2\sin(2x)] = -2 \cdot 2\cos(2x) = -4\cos(2x)dxd[−2sin(2x)]=−2⋅2cos(2x)=−4cos(2x)
- Third derivative: ddx[−4cos(2x)]=4⋅2sin(2x)=8sin(2x)\frac{d}{dx} [-4\cos(2x)] = 4 \cdot 2\sin(2x) = 8\sin(2x)dxd[−4cos(2x)]=4⋅2sin(2x)=8sin(2x)
- Fourth derivative: ddx[8sin(2x)]=8⋅2cos(2x)=16cos(2x)\frac{d}{dx} [8\sin(2x)] = 8 \cdot 2\cos(2x) = 16\cos(2x)dxd[8sin(2x)]=8⋅2cos(2x)=16cos(2x)
Notice that after every 4th derivative, the function repeats in a similar form, with an increasing factor. So, the derivatives of y=cos(2x)y = \cos(2x)y=cos(2x) follow a cyclical pattern with a period of 4. In general:
- The 0th derivative is cos(2x)\cos(2x)cos(2x).
- The 1st derivative is −2sin(2x)-2\sin(2x)−2sin(2x).
- The 2nd derivative is −4cos(2x)-4\cos(2x)−4cos(2x).
- The 3rd derivative is 8sin(2x)8\sin(2x)8sin(2x).
- The 4th derivative is 16cos(2x)16\cos(2x)16cos(2x).
- This pattern repeats every 4 derivatives.
Applying this to the 20th derivative:
Since the pattern repeats every 4 derivatives, we find the remainder when 20 is divided by 4:20÷4=5 remainder 020 \div 4 = 5 \text{ remainder } 020÷4=5 remainder 0
Thus, the 20th derivative corresponds to the 0th derivative in the cycle. The 0th derivative is simply the original function, cos(2x)\cos(2x)cos(2x), multiplied by a constant factor, as seen in the pattern.
Conclusion:
The 20th derivative of y=cos(2x)y = \cos(2x)y=cos(2x) is:f(20)(x)=220cos(2x)f^{(20)}(x) = 2^{20} \cos(2x)f(20)(x)=220cos(2x)
This result comes from the fact that the derivative increases by a factor of 2 with each step in the cycle, and after 20 derivatives (5 full cycles), the constant factor is 2202^{20}220.
