The Constant-Multiple Rule
The power rule handles
x^n. But real functions come with numbers stuck out front, like
5x^2 or \tfrac{1}{2}x^4. What does a
constant multiplier do to the derivative? Almost nothing — it just comes along for the ride:
\frac{d}{dx}\bigl[\,c\cdot f(x)\,\bigr] = c\cdot f'(x)
Scaling the curve scales the slope
Multiplying a function by c stretches its graph vertically by
c. Every rise gets c times taller while
the runs stay the same — so every slope is multiplied by c too.
Drag the stretch factor and watch both the curve and its tangent's steepness scale together.
Why it works
It falls straight out of the
difference quotient:
a constant factor in the top can be pulled right out of the fraction, and out of the limit:
\frac{c\,f(x+h) - c\,f(x)}{h} = c\cdot\frac{f(x+h) - f(x)}{h} \;\xrightarrow{\,h\to 0\,}\; c\cdot f'(x)
So to differentiate c\,x^n, keep the c
and power-rule the x^n:
\frac{d}{dx}\bigl[\,c\,x^n\,\bigr] = c\cdot n\,x^{\,n-1}
For example \dfrac{d}{dx}[5x^2] = 5\cdot 2x = 10x, and
\dfrac{d}{dx}[7x^3] = 7\cdot 3x^2 = 21x^2.
See it together
Here is f(x) = 3x^2 beside its derivative
f'(x) = 6x. The faint curves are the un-scaled
x^2 and 2x — the bold ones are exactly
three times as tall, slope and all.