Calculus: transcendental functions and derivatives

Tutoring calculus, transcendental functions are important. The tutor mentions some examples of derivatives with them.

Transcendental functions are ones that can’t be written in polynomial form. Some examples: sinx, logx, e^x. Importantly, their derivates don’t collapse the way those of polynomials do.

The derivative of sinx, for instance, is cosx. Then, the derivative of cosx is -sinx.

With the product rule, (uv)’ = u’v + uv’, a product involving a transcendental will typically increase in complexity with successive derivates rather than decrease. For instance, the derivative of xsinx, by that same product rule, will be (1)sinx + xcosx.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus. Toronto: D C Heath, 1989.