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

The derivative of sin⁡(1−x)\sin\left(1 - x\right) is −cos⁡(1−x)-\cos\left(1 - x\right).

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

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Graph of f(x) = sin(1 - x) and its derivative-3-2-1123-2-11234slope -1f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(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)=sin⁡(1−x)f(x) = \sin\left(1 - x\right) with respect to x. The rule that applies is the chain rule.

    f(x)=sin⁡(1−x)f(x) = \sin\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 sin(u); its derivative evaluated at g(x), times g'(x):

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

    Simplify:

    f′(x)=−cos⁡(1−x)f'(x) = -\cos\left(1 - x\right)
  5. 5

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

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

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

xf(x)f'(x)
-20.141120010.9899925
-10.909297430.41614684
0.50.47942554-0.87758256
10-1
2-0.84147098-0.54030231
3-0.909297430.41614684

Questions

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

The derivative of sin⁡(1−x)\sin\left(1 - x\right) is −cos⁡(1−x)-\cos\left(1 - x\right).

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

-sin(1 - x).

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

The chain rule.

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

-1.

Related derivatives

    Derivative of sin(1 - x): -cos(1 - x) (with steps) | WordGPT