DIFFERENTIATION 04
The Power Rule
The power rule gives a direct way to differentiate powers of \(x\).
The main idea
For
\[ f(x)=x^n, \]the power rule is
The exponent comes down as a multiplier, and the exponent is reduced by \(1\).
1. A simple example
Consider
\[ y=x^3. \]The exponent is \(3\).
Bring the \(3\) down:
\[ 3x^3. \]Then reduce the exponent by \(1\):
\[ 3x^{3-1}. \]Therefore,
2. Why the rule works
The power rule comes from the definition of the derivative.
For example, let
\[ f(x)=x^3. \]From first principles,
\[ f'(x) = \lim_{h\to0} \frac{(x+h)^3-x^3}{h}. \]Expand:
\[ (x+h)^3 = x^3+3x^2h+3xh^2+h^3. \]Therefore,
\[ f'(x) = \lim_{h\to0} \frac{ 3x^2h+3xh^2+h^3 }{h}. \]For \(h\neq0\),
\[ f'(x) = \lim_{h\to0} \left( 3x^2+3xh+h^2 \right). \]As \(h\to0\),
\[ 3xh\to0 \]and
\[ h^2\to0. \]So
\[ f'(x)=3x^2. \]The power rule allows us to obtain this result directly without repeating the limit calculation every time.
3. More examples
If
\[ y=x^5, \]then
\[ \frac{dy}{dx}=5x^4. \]If
\[ y=x^8, \]then
\[ \frac{dy}{dx}=8x^7. \]4. The derivative of \(x\)
Remember that
\[ x=x^1. \]Using the power rule,
\[ \frac{d}{dx}x = 1x^{1-1}. \]Therefore,
\[ \boxed{ \frac{d}{dx}x=1 } \]because
\[ x^0=1. \]5. The derivative of a constant
A constant does not change as \(x\) changes.
Therefore,
for any constant \(c\).
For example,
\[ \frac{d}{dx}(7)=0. \]6. Constant multiples
If a power is multiplied by a constant, keep the constant and differentiate the power.
For example,
\[ y=4x^3. \]Then
\[ \frac{dy}{dx} = 4(3x^2) = 12x^2. \]In general,
\[ \boxed{ \frac{d}{dx}(cx^n) = cnx^{n-1} } \]7. Differentiating a polynomial
Differentiate each term separately.
For example,
\[ y=3x^4-2x^2+5x-7. \]Differentiate term by term:
\[ \frac{dy}{dx} = 12x^3-4x+5. \]The constant \(-7\) disappears because its derivative is zero.
8. Negative powers
The power rule also works with negative powers.
Consider
\[ y=x^{-2}. \]Then
\[ \frac{dy}{dx} = -2x^{-3}. \]Equivalently,
\[ \boxed{ \frac{d}{dx} \left(\frac{1}{x^2}\right) = -\frac{2}{x^3} } \]9. Fractional powers
Roots can be written as fractional powers.
For example,
\[ \sqrt{x}=x^{1/2}. \]Using the power rule,
\[ \frac{d}{dx}x^{1/2} = \frac12x^{-1/2}. \]Therefore,
for \(x>0\).
10. Example with a fractional power
Consider
\[ y=x^{3/2}. \]Then
\[ \frac{dy}{dx} = \frac32x^{1/2}. \]So
\[ \boxed{ \frac{dy}{dx} = \frac32\sqrt{x} } \]11. What the derivative tells us
Suppose
\[ y=x^3. \]Then
\[ \frac{dy}{dx}=3x^2. \]At \(x=2\),
\[ \left. \frac{dy}{dx} \right|_{x=2} = 3(2)^2 = 12. \]This means that the graph of \(y=x^3\) has tangent gradient \(12\) at \(x=2\).
12. The pattern
For example,
\[ x^6 \longrightarrow 6x^5, \] \[ x^{-3} \longrightarrow -3x^{-4}, \] \[ x^{1/2} \longrightarrow \frac12x^{-1/2}. \]13. The important point
The power rule is not a separate definition of differentiation.
The derivative is still defined by the limit
\[ f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}. \]The power rule is a result derived from that definition which makes differentiation much faster.