Find the derivatives of the following trigonometric functions: 1) f(x) = sin(3x) 2) f(x) = cos(5x) 3) f(x) = cot(8x) 4) g(x) = sec(8x) 5) P(x) = tan(2x) 6) h(x) = cosec(6x)

Steps to Solve

1. Derivative of $sin(ax)$

The derivative of $sin(ax)$ is $a \cdot cos(ax)$. Therefore, the derivative of $f(x) = sin(3x)$ is $f'(x) = 3cos(3x)$. The provided answer is $cos(3x)$ - missing the factor of 3. This is incorrect.

2. Derivative of $cos(ax)$

The derivative of $cos(ax)$ is $-a \cdot sin(ax)$. Therefore, the derivative of $f(x) = cos(5x)$ is $f'(x) = -5sin(5x)$. The provided answer is $-sin(5x)$ - missing the factor of 5. This is incorrect.

3. Derivative of $cot(ax)$

The derivative of $cot(ax)$ is $-a \cdot cosec^2(ax)$. Thus, the derivative of $f(x) = cot(8x)$ is $f'(x) = -8cosec^2(8x)$. The provided answer is $-cosec^2(8x)$ - missing the factor of 8. This is incorrect.

4. Derivative of $sec(ax)$

The derivative of $sec(ax)$ is $a \cdot sec(ax)tan(ax)$. Therefore, the derivative of $g(x) = sec(8x)$ is $g'(x) = 8sec(8x)tan(8x)$. The provided answer, $\frac{sec(x)8x}{tan(x)}$, is incorrect. It should be $8sec(8x)tan(8x)$.

5. Derivative of $tan(ax)$

The derivative of $tan(ax)$ is $a \cdot sec^2(ax)$. Therefore, the derivative of $p(x) = tan(2x)$ is $p'(x) = 2sec^2(2x)$. The provided answer $\frac{1}{1+x^2}2x$ which simplifies to $\frac{2x}{1+x^2}$ which is incorrect. It looks like someone was trying to apply the derivative for $arctan(x)$ mistakenly.

6. Derivative of $cosec(ax)$

The derivative of $cosec(ax)$ is $-a \cdot cosec(ax)cot(ax)$. The derivative shown is incomplete lacking both the factor of 6, as well as the $cot(ax)$ term.