Calculus Lessons: How to Solve a Calculus Problem

A calculus problem usually asks one of two things: how fast something is changing (a derivative) or how much has built up (an integral). These lessons show how to recognise which one you need and work it out step by step.

Derivative or integral?

“Rate of change”, “gradient of the curve”, “velocity from position”, “maximum or minimum” → differentiate.
“Area under the curve”, “total amount”, “position from velocity” → integrate.

Two short calculus problems

Slope of $y = x^3 - 3x$ at $x = 2$ : the derivative is $3x^2 - 3$ , so the slope is $3(4) - 3 = 9$ .

Area under $y = 3x^2$ from 0 to 2: $\int_0^2 3x^2\,dx = \left[x^3\right]_0^2 = 8$ .

How to work through a calculus problem

Write down what is given and what is asked, with units.
Decide: derivative or integral (or both).
Simplify the function first — expand brackets, write roots as powers.
Apply the rule, then substitute the numbers.
Sense-check: is the sign and size of the answer reasonable?

Frequently asked questions

How do I start a calculus problem?

Decide whether it asks about a rate of change (derivative) or an accumulated total (integral), then simplify the function before applying the rule.

What should I know before learning calculus?

Algebra — especially powers, brackets and rearranging — plus graphs of functions. Most calculus mistakes are really algebra slips.

Can I check a derivative or integral?

Yes. Differentiate your integral to see if you get the original function back, or use the Sumzo derivative and integral calculators, which show each step.