The derivative of sin cube x is equal to 3sin^{2}x cosx. Here we learn how to differentiate sin^3x. The sin^3x derivative formula is d/dx(sin^{3}x) = 3sin^{2}x cosx.

The derivative of sine cube x is denoted by d/dx(sin^{3}x) 0r (sin^{3}x)$’$ and it is equal to

- d/dx(sin
^{3}x) = 3sin^{2}x cos x. - (sin
^{3}x)$’$ = 3sin^{2}x cos x.

## Derivative of sin^{3}x by Chain Rule

To find the derivative of sine cube x by the chain rule of derivatives, let us put

z=sinx

∴ $\dfrac{dz}{dx}$ = cosx

Now, by the chain rule, the derivative of sin^{3}x is equal to

$\dfrac{d}{dx}\big(\sin^3 x \big)$ = $\dfrac{d}{dz}(z^3)$ × $\dfrac{dz}{dx}$

= 3z^{2} × cosx by the power rule of derivatives and $\frac{dz}{dx}$ =cosx determined above.

= 3z^{2} cosx

= 3 sin^{2}x cosx as z=sinx.

Therefore, the derivative of sin^{3}x is equal to 3 sin^{2}x cosx and this is obtained by the chain rule of derivatives.

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## Derivative of sin^{3}x by Product Rule

Note that sin^{3}x can be written as a product of sinx and sin^{2}x. So by the product rule, the derivative of sin cube x will be computed as follows:

$\dfrac{d}{dx}\big(\sin^3 x \big)$ = $\dfrac{d}{dx}\big(\sin x \cdot \sin^2 x \big)$

= $\sin x\dfrac{d}{dx}\big(\sin^2 x \big)$ + $\sin^2 x\dfrac{d}{dx}\big(\sin x \big)$

= sinx × 2sinx cosx + sin^{2}x cosx

= 2sin^{2}x cosx + sin^{2}x cosx

= 3sin^{2}x cosx.

So the differentiation of sin^{3}x is equal to 3sin^{2}x cosx and this is obtained by the product rule of differentiation.

## FAQs

**Q1: What is the derivative of sin^3x?**

Answer: The derivative of sin^{3}x is 3cosx sin^{2}x.

**Q2: Write the derivative formula of sin^3x.**

Answer: The derivative formula of sin^3x is given by d/dx(sin^{3}x) = 3cosx sin^{2}x.

**Q3: If y=sin^3x, then find dy/dx.**

Answer: If y=sin^{3}x, then dy/dx = 3sin^{2}x cosx.