The derivative of sin^{5}x is equal to 5sin^{4}x cosx. In this post, we will learn how to differentiate sin^{5}x, that is, sine to the power 5 of x.

## Derivative of sin^{5}x Formula:

The derivative of sin^{5}x is denoted by $\dfrac{d}{dx}(\sin^5 x)$ or $(\sin^5 x)’$. The formula of the derivative of sin^{5}x is given below:

$\dfrac{d}{dx}(\sin^5 x)=5\sin^4 x \cos x$, or $(\sin^5 x)’$=5sin^{4}x cosx.

Derivative of sin5x

**Question:** What is the Derivative of sin^{5}x?

*Answer:* The Derivative of sin^{5}x is 5sin^{4}x cosx.

**Proof: **

**Step 1:** Let us assume that z=sin x. Then we can write our function as

sin^{5}x=z^{5}

**Step 2:** Note that $\dfrac{dz}{dx}=\cos x$ as $z=\sin x$.

**Step 3:** By the chain rule, the derivative of $\sin^5x$ will be equal to

$\dfrac{d}{dx}(\sin^5x)=\dfrac{d}{dz}(z^5) \cdot \dfrac{dz}{dx}$

$=5z^4 \cdot (\cos x)$ by the power rule of derivatives: $\dfrac{d}{dx}(x^n)=nx^{n-1}$

$=5\sin^4 x \cdot \cos x$ as $z=\sin x$

$=5\sin^4 x \cos x$.

**Conclusion:** The derivative of sin^{5}x is 5sin^{4}x cosx and this is obtained by the chain rule and the power rule of derivatives.

In a similar way, one can obtain the derivative of sin^{n} x, which is given below:

$\dfrac{d}{dx}(\sin^n x)=n\sin^{n-1} x \cos x$.

For example,

- The derivative of sin
^{2}x is 2sin x cos x. - The derivative of sin
^{3}x is 3sin^{2}x cos x. - The derivative of sin
^{4}x is 4sin^{3}x cos x.

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## FAQs

**Q1: What is the derivative of sin ^{5}x?**

**Answer:** The derivative of sin^{5}x is equal to 5sin^{4}x cosx.