First principles on $x^2$ gives $2x$; on $x^3$ it gives $3x^2$; on $x^4$, $4x^3$. The pattern is the power rule: $\dfrac{d}{dx}\,x^n = n\,x^{n-1}$, read aloud: the derivative of x to the n is n times x to the n minus one. Here $n$ is the exponent, any fixed real number, and the rule does two things in one move: the old exponent comes down in front as a factor, and the exponent itself drops by one. Both or neither: $x^3$ becomes $3x^2$, not $x^2$ (lowered, never brought down) and not $3x^3$ (brought down, never lowered). A coefficient in front rides along: $\dfrac{d}{dx}\,5x^3 = 15x^2$. And $x$ itself is $x^1$, whose derivative is $1 \cdot x^0 = 1$: a line of slope 1.

A polynomial is differentiated one term at a time, the derivative of a sum being the sum of the derivatives. A constant term is a flat line, and its derivative is 0, not the constant: $f(x) = 2x^3 - 5x + 7$ has $f'(x) = 6x^2 - 5$, and the 7 is gone. Keeping it, writing $6x^2 - 5 + 7$, is the second trap of the lesson. At a point, substitute after differentiating: $f'(2) = 6(4) - 5 = 19$. As a check against the definition, the slope of $x^3$ at $x = 4$ is $3 \cdot 4^2 = 48$, and the secant from 3.999 to 4.001 has slope 48.000001.