Derivatives as limits and local rates | Code Scholars Tutoring

Understand the idea

The derivative is the limit of secant slopes when that limit exists. It is both the slope of the tangent and an instantaneous rate with output-units per input-unit. Estimate from nearby table entries; a symmetric secant is often useful but still approximate. Distinguish a derivative value at one point from a derivative function on a domain.

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h

Worked example

Use the definition to find the derivative of $\displaystyle f(x)=x^2$ at $\displaystyle x=3$ .

The difference quotient is

\displaystyle ((3+h)^2-9)/h=(6h+h^2)/h=6+h

for

\displaystyle h\ne0

. Taking

\displaystyle h\to0

gives

\displaystyle f'(3)=6

. The tangent slope at

\displaystyle (3,9)

is

\displaystyle 6

.

Try it yourself

If $\displaystyle f(1.9)=5.61$ and $\displaystyle f(2.1)=6.41$ , estimate $\displaystyle f'(2)$ .

Show solution and reasoning

The symmetric secant estimate is $\displaystyle (6.41-5.61)/(2.1-1.9)=4$ . It is an approximation, not an exact consequence of two data values.