What Is the Derivative of cos(x) with Respect to 1/x?

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SUMMARY

The derivative of cos(x) with respect to 1/x can be calculated using the chain rule. By substituting y = 1/x, the derivative becomes (d/dy) cos(1/y). This leads to the expression -sin(1/y) * (-1/y^2), resulting in sin(1/y)/y^2. This method effectively demonstrates the application of the chain rule in differentiating trigonometric functions with respect to a reciprocal variable.

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  • Understanding of basic calculus concepts, particularly derivatives.
  • Familiarity with the chain rule in differentiation.
  • Knowledge of trigonometric functions and their derivatives.
  • Ability to manipulate algebraic expressions involving variables.
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  • Explore derivatives of other trigonometric functions with respect to reciprocal variables.
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Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of differentiation techniques involving trigonometric functions and reciprocal variables.

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if (d/dx) cos(x) = -sin(x) then (d/d{1/x}) cos(x) = ? i.e. the derivative of cos(x) with respect to 1/x
 
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You can subsitiute ##\ y = {1\over x}\ ## and use the chain rule to determine the derivative of ##\cos\left ( 1\over y \right ) ##.
 

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