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Derivative of ln⁡(x2)\ln\left(x^{2}\right)

The derivative of ln⁡(x2)\ln\left(x^{2}\right) is 2xx2\frac{2 x}{x^{2}}.

f(x) =
ddx[ln⁡(x2)]=2xx2\frac{d}{dx}\left[\ln\left(x^{2}\right)\right] = \frac{2 x}{x^{2}}

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Graph of f(x) = ln(x^2) and its derivative-3-2-1123-8-6-4-224slope 2f(x)f'(x)tangent at x = 1
Graph of f(x) = ln(x^2) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.

How to work it out

  1. 1

    We differentiate f(x)=ln⁡(x2)f(x) = \ln\left(x^{2}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=ln⁡(x2)f(x) = \ln\left(x^{2}\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)=x2g(x) = x^{2}.

    g(x)=x2,g′(x)=2xg(x) = x^{2},\quad g'(x) = 2 x
  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)=2xx2f'(x) = \frac{2 x}{x^{2}}
  5. 5

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

    f′′(x)=2x2−2x2x(x2)2f''(x) = \frac{2 x^{2} - 2 x 2 x}{\left(x^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\ln\left(x^{2}\right)\right] = \frac{2 x}{x^{2}} \]
\[ f(x) = \ln\left(x^{2}\right) \]
\[ g(x) = x^{2},\quad g'(x) = 2 x \]
\[ f'(x) = \left[\frac{1}{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{2 x}{x^{2}} \]
\[ f''(x) = \frac{2 x^{2} - 2 x 2 x}{\left(x^{2}\right)^{2}} \]

Values of f and f' for ln(x2x^{2})

xf(x)f'(x)
-21.3862944-1
-10-2
0.5-1.38629444
102
21.38629441
32.19722460.66666667

Questions

What is the derivative of ln(x2x^{2})?

The derivative of ln⁡(x2)\ln\left(x^{2}\right) is 2xx2\frac{2 x}{x^{2}}.

What is the second derivative of ln(x2x^{2})?

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

Which rule do you use for ln(x2x^{2})?

The chain rule.

What is the slope of ln(x2x^{2}) at x = 1?

2.

Related derivatives

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