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Exercise

Find the differential dy of y=cos(x)

Step-by-step Solution

1

Math interpretation of the question

$\frac{d}{dx}\left(y=\cos\left(x\right)\right)$
2

Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(\cos\left(x\right)\right)$
3

The derivative of the linear function is equal to $1$

$dy=\frac{d}{dx}\left(\cos\left(x\right)\right)$
4

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$dy=-\frac{d}{dx}\left(x\right)\sin\left(x\right)$
5

The derivative of the linear function is equal to $1$

$dy=-\sin\left(x\right)$

Final answer to the exercise

$dy=-\sin\left(x\right)$

Try other ways to solve this exercise

  • Choose an option
  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Load more...
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$\frac{d}{dx}\left(x^2+xy-y^2=4\right)$

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Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

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