"Understand and practice using the definition of a derivative" means you should be able to use the limit definition of the derivative to find derivatives, not just use shortcut rules.
The definition is:
[
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
]
Here's how to practice it.
-
Memorize the definition.
Know the formula and what each part means:
- (f(x+h)): the function with (x) replaced by (x+h)
- (f(x)): the original function
- Divide by (h)
- Simplify
- Take the limit as (h\to0)
-
Work through simple examples.
For example, if (f(x)=x^2):
[
\text{Step 1: }f(x+h)=(x+h)^2=x^2+2xh+h^2
]
[
\text{Step 2: }\frac{f(x+h)-f(x)}{h}
\frac{x^2+2xh+h^2-x^2}{h}
]
[
\frac{2xh+h^2}{h}
=2x+h
]
[
\text{Step 3: }\lim_{h\to0}(2x+h)=2x
]
So,
[
f'(x)=2x.
]
- Practice with different functions. Try:
- (f(x)=3x+1)
- (f(x)=x^2)
- (f(x)=x^3)
- (f(x)=\frac{1}{x})
- (f(x)=\sqrt{x})
- Know what the derivative represents.
- The instantaneous rate of change of a function.
- The slope of the tangent line at a point.
- Check your work.
After finding the derivative using the definition, compare it with the derivative you get using differentiation rules (like the power rule). They should match.
A good practice progression is:
- Linear functions
- Quadratic functions
- Cubic functions
- Rational functions
- Radical functions
If you're studying for a calculus class, being able to compute derivatives from the definition for the first two or three types is usually expected before moving on to the shortcut differentiation rules.
