How does the derivative tell me where a function is rising or falling (useful for graphs and trends)?
The sign of the first derivative gives the direction.
The sign of the first derivative gives the direction. Where f'(x) > 0 the function is increasing (going uphill); where f'(x) < 0 it's decreasing (downhill); and f'(x) = 0 marks the turning points between the two. So to sketch a curve or find when a profit/temperature is rising, you just solve f'(x) = 0 to locate the turning points and check the sign of f' on each interval.
Key point
f'(x) > 0 → increasing, f'(x) < 0 → decreasing, f'(x) = 0 → turning point.
Common mistake
reading the value of f instead of the sign of f' to judge rising/falling.
Memory tip
the slope's sign (f') tells you up or down, not the height f.
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