Derivatives of Polynomials by First Principles - Example 1

In this video, I demonstrate how to find the derivative of f(x) = x² - 8x + 2 using by First Principles.

The derivative is defined as:

\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}

The first step is to expand and simplify the term f(x+h)...

\begin{align*} f(x+h) &= (x+h)^2 - 8(x+h) + 2\\ &= x^2 + 2xh + h^2 - 8x - 8h + 2 \end{align*}

With the numerator being f(x+h)-f(x), this equates to...

\begin{align*} f(x+h)-f(x) &= (x+h)^2 - 8(x+h) + 2-\left ( x^2-8x+2 \right )\\ &= x^2 + 2xh + h^2 - 8x - 8h + 2 - x^2 + 8x - 2\\ &= 2xh + h^2 - 8h \end{align*}

The [f(x+h)-f(x)]/h equates to...

\frac{f(x+h)-f(x)}{h}=\frac{2xh + h^2 - 8h}{h}=2x+h-8