1. Home
  2. >Differential Calculus
  3. >Differentiation
Help

Power rule

All in One Place

Everything you need for JC, LC, and college level maths and science classes.

Learn with Ease

We’ve mastered the national curriculum so that you can revise with confidence.

Instant Help

24/7 access to the best tips, walkthroughs, and practice exercises available.

0/12
?
Examples
Lessons
  1. power rule: ddx(xn)=n  xn−1\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}
    1.   ddx(x5){\;}\frac{{d}}{{{d}x}}\left( {{x^5}} \right)
    2.   ddx(x){\;}\frac{{d}}{{{d}x}}\left( x \right)
    3.   ddx(3){\;}\frac{{d}}{{{d}x}}\left( 3 \right)
  2. constant multiple rule: ddx[cf(x)]=c  ddxf(x)\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)
    1.   ddx(4x3){\;}\frac{{d}}{{{d}x}}\left( {4{x^3}} \right)
    2.   ddx(6x){\;}\frac{{d}}{{{d}x}}\left( {6x} \right)
    3.   ddx(−x){\;}\frac{{d}}{{{d}x}}\left( { - x} \right)
  3. ddx(x10−5x7+13x4−20x3+x2−8x−1000)\frac{{d}}{{{d}x}}\left( {{x^{10}} - 5{x^7} + \frac{1}{3}{x^4} - 20{x^3} + {x^2} - 8x - 1000} \right)

    sum rule: ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)\frac{{d}}{{{d}x}}\left[ {f\left( x \right) + g\left( x \right)} \right] = \frac{{d}}{{{d}x}}f\left( x \right) + \frac{{d}}{{{d}x}}g\left( x \right)
    difference rule: ddx[f(x)−g(x)]=ddxf(x)−ddxg(x)\frac{{d}}{{{d}x}}\left[ {f\left( x \right) - g\left( x \right)} \right] = \frac{{d}}{{{d}x}}f\left( x \right) - \frac{{d}}{{{d}x}}g\left( x \right)
    1. negative exponents: 1x=x−1\frac{1}{x} = {x^{ - 1}} and 1xn=x−n\frac{1}{{{x^n}}} = {x^{ - n}}
      1.   ddx(1x2){\;}\frac{{d}}{{{d}x}}\left( {\frac{1}{{{x^2}}}} \right)
      2.   ddx(−53x){\;}\frac{{d}}{{{d}x}}\left( {\frac{{ - 5}}{{3x}}} \right)
    2. rational exponents: x=x12\sqrt x = {x^{\frac{1}{2}}} and bxa=xab{^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}
      1.   ddx(3x5){\;}\frac{{d}}{{{d}x}}\left( {{^3}\sqrt{{{x^5}}}} \right)
      2.   ddx(x){\;}\frac{{d}}{{{d}x}}\left( {\sqrt x } \right)
      3.   ddx(821x3){\;}\frac{{d}}{{{d}x}}\left( {\frac{8}{{21\sqrt {{x^3}} }}} \right)
    Topic Notes
    ?
    When using the Definition of Derivative, finding the derivative of a long polynomial function with large exponents, or powers, can be very demanding. To avoid this, we introduce you one of the most powerful differentiation tools that simplifies this entire differentiation process – the Power Rule. In this section, we will see how the Power Rule allows us to easily derive the slope of a polynomial function at any given point.
    POWER RULE: ddx(xn)=n  xn−1\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}} , where nn is any real number