Derivative of cos cube x | Cos^3x Derivative

The derivative of cos cube x (cos^3x) is equal to -3cos²x sinx. In this post, we will find the derivative of cos cube x using the chain rule.

The derivative formula of cos³x is given as follows:

d/dx (cos³x) = -3cos²x sinx.

Derivative of Cos^3x

The derivative of cos3x is -3cos2x sinx.

Explanation:

As cos³x is a composite function, we will use the chain rule to find its derivative. To do so, let us put

u = cosx.

Thus, cos³x = u³.

As u=cosx, we have that du/dx = -sinx.

Now, by the chain rule the derivative of cos cube x will be equal to

$\dfrac{d}{dx}$ (cos³x) = $\dfrac{d}{dx}$ (u³)

= $\dfrac{d}{du}$ (u³) × $\dfrac{du}{dx}$

= 3u² × (-sinx) by the power rule of derivatives: d/dx (xⁿ) =nx^n-1.

= – 3cos²x sinx.

So the derivative of cos³x (cos cube x) is equal to – 3cos²x sinx, that is, d/dx (cos³x) = -3cos²x sinx which is proved by the chain rule of derivatives.

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