Power rule

Power rule

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.

Lessons

POWER RULE: ddx(xn)=nxn−1\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}} , where nn is any real number
  • 1.
    power rule: ddx(xn)=nxn−1\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}
    a)
    ddx(x5){\;}\frac{{d}}{{{d}x}}\left( {{x^5}} \right)

    b)
    ddx(x){\;}\frac{{d}}{{{d}x}}\left( x \right)

    c)
    ddx(3){\;}\frac{{d}}{{{d}x}}\left( 3 \right)

  • 2.
    constant multiple rule: ddx[cf(x)]=cddxf(x)\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)
    a)
    ddx(4x3){\;}\frac{{d}}{{{d}x}}\left( {4{x^3}} \right)

    b)
    ddx(6x){\;}\frac{{d}}{{{d}x}}\left( {6x} \right)

    c)
    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)
  • 4.
    negative exponents: 1x=x−1\frac{1}{x} = {x^{ - 1}} and 1xn=x−n\frac{1}{{{x^n}}} = {x^{ - n}}
    a)
    ddx(1x2){\;}\frac{{d}}{{{d}x}}\left( {\frac{1}{{{x^2}}}} \right)

    b)
    ddx(−53x){\;}\frac{{d}}{{{d}x}}\left( {\frac{{ - 5}}{{3x}}} \right)

  • 5.
    rational exponents: x=x12\sqrt x = {x^{\frac{1}{2}}} and bxa=xab{^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}
    a)
    ddx(3x5){\;}\frac{{d}}{{{d}x}}\left( {{^3}\sqrt{{{x^5}}}} \right)

    b)
    ddx(x){\;}\frac{{d}}{{{d}x}}\left( {\sqrt x } \right)

    c)
    ddx(821x3){\;}\frac{{d}}{{{d}x}}\left( {\frac{8}{{21\sqrt {{x^3}} }}} \right)

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39.
Differentiation
39.1
Power rule
39.2
Slope and equation of tangent line
39.3
Chain rule
39.4
Derivative of trigonometric functions
39.5
Derivative of exponential functions
39.6
Product rule
39.7
Quotient rule
39.8
Derivative of logarithmic functions
39.9
Higher order derivatives
39.10
Rectilinear Motion: Derivative
39.11
Critical number & maximum and minimum values
GCE O-Level A Maths
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