1. Home
  2. >Calculus
  3. >Derivatives
Help

Power rule

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

Get the most by viewing this topic in your current grade. Pick your course now.

?
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