Derivative using linearity rules

Apply the sum/difference rule to differentiate each term separately: $f'(x) = \frac{d}{dx}[5\tan x] - \frac{d}{dx}[2x^3] + \frac{d}{dx}[6]$. For the first term, use the constant multiple rule with the known trig derivative $\frac{d}{dx}[\tan x] = \sec^2 x$, giving $5\sec^2 x$.

For the second term, apply the constant multiple rule together with the power rule: $\frac{d}{dx}[2x^3] = 2\cdot 3x^2 = 6x^2$. The constant term $6$ has derivative $0$ by the constant rule. Combining all pieces gives $f'(x) = 5\sec^2 x - 6x^2$.