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Derivative of ln⁡(x3)\ln\left(x^{3}\right)

The derivative of ln⁡(x3)\ln\left(x^{3}\right) is 3x2x3\frac{3 x^{2}}{x^{3}}.

f(x) =
ddx[ln⁡(x3)]=3x2x3\frac{d}{dx}\left[\ln\left(x^{3}\right)\right] = \frac{3 x^{2}}{x^{3}}

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Graph of f(x) = ln(x^3) and its derivative-3-2-1123-10-55slope 3f(x)f'(x)tangent at x = 1
Graph of f(x) = ln(x^3) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.

How to work it out

  1. 1

    We differentiate f(x)=ln⁡(x3)f(x) = \ln\left(x^{3}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=ln⁡(x3)f(x) = \ln\left(x^{3}\right)
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=x3g(x) = x^{3}.

    g(x)=x3,g′(x)=3x2g(x) = x^{3},\quad g'(x) = 3 x^{2}
  3. 3

    The outer function is ln(u); its derivative evaluated at g(x), times g'(x):

    f′(x)=[1u]u=g(x)⋅g′(x)f'(x) = \left[\frac{1}{u}\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    f′(x)=3x2x3f'(x) = \frac{3 x^{2}}{x^{3}}
  5. 5

    Differentiating once more gives the second derivative, which tells you about curvature:

    f′′(x)=6xx3−3x23x2(x3)2f''(x) = \frac{6 x x^{3} - 3 x^{2} 3 x^{2}}{\left(x^{3}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\ln\left(x^{3}\right)\right] = \frac{3 x^{2}}{x^{3}} \]
\[ f(x) = \ln\left(x^{3}\right) \]
\[ g(x) = x^{3},\quad g'(x) = 3 x^{2} \]
\[ f'(x) = \left[\frac{1}{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{3 x^{2}}{x^{3}} \]
\[ f''(x) = \frac{6 x x^{3} - 3 x^{2} 3 x^{2}}{\left(x^{3}\right)^{2}} \]

Values of f and f' for ln(x3x^{3})

xf(x)f'(x)
0.5-2.07944156
103
22.07944151.5
33.29583691

Questions

What is the derivative of ln(x3x^{3})?

The derivative of ln⁡(x3)\ln\left(x^{3}\right) is 3x2x3\frac{3 x^{2}}{x^{3}}.

What is the second derivative of ln(x3x^{3})?

(6x * x3x^{3} - 3 * x2x^{2} * 3 * x2x^{2}) / (x3x^{3})^2.

Which rule do you use for ln(x3x^{3})?

The chain rule.

What is the slope of ln(x3x^{3}) at x = 1?

3.

Related derivatives

    Derivative of ln(x^3): 3 * x^2 / x^3 (with steps) | WordGPT