Derivative of a sum of trig functions and a linear term

Apply the sum/difference rule to break $h(x)$ into three pieces: $10\sec(x)$, $4\cot(x)$, and $-x$, differentiating each separately. Using the constant multiple rule, the derivative of $10\sec(x)$ is $10$ times the derivative of $\sec(x)$, which is $\sec(x)\tan(x)$, giving $10\sec(x)\tan(x)$. Similarly, the derivative of $4\cot(x)$ is $4$ times the derivative of $\cot(x)$, which is $-\csc^2(x)$, giving $-4\csc^2(x)$.

The derivative of $-x$ is simply $-1$, since $\frac{d}{dx}[x] = 1$. Combining all three results with the appropriate signs from the difference rule gives the final derivative $h'(x) = 10\sec(x)\tan(x) - 4\csc^2(x) - 1$.

The term came from differentiating four cotangent of x. Since the derivative of cotangent is negative cosecant squared, multiplying by the constant four keeps the sign negative. So the cosecant squared term in h prime of x is negative, appearing as negative four cosecant squared x.