Find the derivative of the following functions 'ab initio', that is, using the definition.

$2x^3$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$x^4$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$x^2+1$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$\frac{1}{x}$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$\frac{1}{3x+2}$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$\frac{1}{x^2}$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$\frac{x}{x+1}$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$t\left(t-1\right)$

Find the derivative of the following functions 'ab initio', that is, using the definition.

$s^2-6s+5$