The derivative of sinx cosx is equal to cos2x. Here we learn to find the differentiation of sinx cosx by the product rule.

The derivative formula of the product sinx cosx is given as follows: $\dfrac{d}{dx}$(sinx cosx) = cos2x.

## Find the Derivative of sinx cosx

We know that sin2x = 2 sinx cosx. Thus, it follows that

sinx cosx = $\dfrac{\sin 2x}{2}$.

Differentiating both sides with respect to x, we get that

$\dfrac{d}{dx}(\sin x \cos x$ = $\dfrac{d}{dx}(\dfrac{\sin 2x}{2})$

= $\dfrac{1}{2}\dfrac{d}{dx}(\sin 2x)$

= $\dfrac{1}{2} \times 2 \cos 2x$ as the derivative of sinmx is mcosmx.

= cos2x.

Therefore, the derivative of sinx cosx is cos 2x.

Related Derivatives:

Derivative of cos3x: The derivative of cos3x is -3sin3x. |

Derivative of tan2x: The derivative of tan2x is 2sec^{2}2x. |

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

Derivative of 1/logx: The derivative of 1/log x is -1/x(log x)^{2}. |

## Derivative of sinx cosx by the product rule

As the function sinx cosx is a product of two functions, we can use the product rule to find its derivative.

Note that

$\dfrac{d}{dx}(\sin x \cos x)$ = $\sin x\dfrac{d}{dx}(\cos x)$ + $\cos x\dfrac{d}{dx}(\sin x)$

= $\sin x (-\sin x)$ + $\cos x \cos x$

= – sin^{2}x + cos^{2}x

= cos2x using the trigonometric formula cos^{2}x – sin^{2}x = cos2x.

So the derivative of sinx cosx is equal to cos2x and this is obtained by the product rule of derivatives.

## FAQs

**Q1: What is the derivative of sinx cosx?**

Answer: The derivative of sinx cosx is cos2x.

**Q2: If y=sinx cosx, then find dy/dx?**

Answer: If y=sinx cosx, then dy/dx = cos2x.