Closed formulas for computing higher-order derivatives of functions involving exponential functions

For integers k ≥ 1 and n   ≥ 0, the functions 1/k(1−λeαt)1/(1−λeαt)k and the derivatives (1/(1−λeαt))(n)(1/(1−λeαt))(n) can be expressed each other by linear combinations. Based on this viewpoint, we find several new closed formulas for higher-order derivatives of trigonometric and hyperbolic functions, derive a higher-order convolution formula for the tangent numbers, and generalize a recurrence relation for the tangent numbers.