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Derivative of tan⁡(1−x)\tan\left(1 - x\right)

The derivative of tan⁡(1−x)\tan\left(1 - x\right) is −1cos⁡(1−x)2\frac{-1}{\cos\left(1 - x\right)^{2}}.

f(x) =
ddx[tan⁡(1−x)]=−1cos⁡(1−x)2\frac{d}{dx}\left[\tan\left(1 - x\right)\right] = \frac{-1}{\cos\left(1 - x\right)^{2}}

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Graph of f(x) = tan(1 - x) and its derivative-3-2-1123-60-50-40-30-20-1010slope -1f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(1 - x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.

How to work it out

  1. 1

    We differentiate f(x)=tan⁡(1−x)f(x) = \tan\left(1 - x\right) with respect to x. The rule that applies is the chain rule.

    f(x)=tan⁡(1−x)f(x) = \tan\left(1 - x\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)=1−xg(x) = 1 - x.

    g(x)=1−x,g′(x)=−1g(x) = 1 - x,\quad g'(x) = -1
  3. 3

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

    f′(x)=[1cos⁡(u)2]u=g(x)⋅g′(x)f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    f′(x)=−1cos⁡(1−x)2f'(x) = \frac{-1}{\cos\left(1 - x\right)^{2}}
  5. 5

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

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

Values of f and f' for tan(1 - x)

xf(x)f'(x)
-2-0.14254654-1.0203195
-1-2.1850399-5.7743992
0.50.54630249-1.2984464
10-1
2-1.5574077-3.4255188
32.1850399-5.7743992

Questions

What is the derivative of tan(1 - x)?

The derivative of tan⁡(1−x)\tan\left(1 - x\right) is −1cos⁡(1−x)2\frac{-1}{\cos\left(1 - x\right)^{2}}.

What is the second derivative of tan(1 - x)?

2cos(1 - x) * sin(1 - x) / (cos(1 - x)^2)^2.

Which rule do you use for tan(1 - x)?

The chain rule.

What is the slope of tan(1 - x) at x = 1?

-1.

Related derivatives

    Derivative of tan(1 - x): -1 / cos(1 - x)^2 (with steps) | WordGPT