Derivative of sin^5 x [by Chain Rule] | sin^5x Derivative

The derivative of sin⁵x is equal to 5sin⁴x cosx. In this post, we will learn how to differentiate sin⁵x, that is, sine to the power 5 of x.

Derivative of sin⁵x Formula:

The derivative of sin⁵x is denoted by ${\displaystyle \frac{d}{dx}}({\sin }^{5}x)$ or $({\sin }^{5}x{)}^{\prime }$. The formula of the derivative of sin⁵x is given below:

${\displaystyle \frac{d}{dx}}({\sin }^{5}x)=5{\sin }^{4}x\cos x$, or $({\sin }^{5}x{)}^{\prime }$=5sin⁴x cosx.

Derivative of sin^5x

Answer: The Derivative of sin5x is 5sin4x cosx.

Proof:

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

sin⁵x=z⁵

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

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

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

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

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

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

Conclusion: The derivative of sin⁵x is 5sin⁴x 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 sinⁿ x, which is given below:

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

For example,

1. The derivative of sin²x is 2sin x cos x.
2. The derivative of sin³x is 3sin² x cos x.
3. The derivative of sin⁴x is 4sin³ x cos x.

Also Read:

Derivative of cos² x

Derivative of sin² x

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