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Derivative of sin⁡(x)cos⁡(x)\frac{\sin\left(x\right)}{\cos\left(x\right)}

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

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

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Graph of f(x) = sin(x) / cos(x) and its derivative-3-2-1123-20204060slope 3.426f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / cos(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.4255.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x)cos⁡(x)f(x) = \frac{\sin\left(x\right)}{\cos\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=sin⁡(x)cos⁡(x)f(x) = \frac{\sin\left(x\right)}{\cos\left(x\right)}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=sin⁡(x)u = \sin\left(x\right) and v=cos⁡(x)v = \cos\left(x\right).

    u=sin⁡(x),v=cos⁡(x)u = \sin\left(x\right),\quad v = \cos\left(x\right)
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=−sin⁡(x)u' = \cos\left(x\right),\quad v' = -\sin\left(x\right)
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=cos⁡(x)cos⁡(x)−sin⁡(x)−sin⁡(x)cos⁡(x)2f'(x) = \frac{\cos\left(x\right) \cos\left(x\right) - \sin\left(x\right) -\sin\left(x\right)}{\cos\left(x\right)^{2}}
  6. 6

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

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

Values of f and f' for sin(x) / cos(x)

xf(x)f'(x)
-22.18503995.7743992
-1-1.55740773.4255188
0.50.546302491.2984464
11.55740773.4255188
2-2.18503995.7743992
3-0.142546541.0203195

Questions

What is the derivative of sin(x) / cos(x)?

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

What is the second derivative of sin(x) / cos(x)?

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

Which rule do you use for sin(x) / cos(x)?

The quotient rule.

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

3.4255188.

Related derivatives

    Derivative of sin(x) / cos(x): (cos(x) * cos(x) - sin(x) * -sin(x)) / cos(x)^2 (with steps) | WordGPT