Derivative of sin(x)*cos(x)

Step-by-step solution

1. 1Product Rule
   $$\frac{d}{dx}\left[\sin \left(x\right)\cdot \cos \left(x\right)\right]=\frac{d}{dx}\left[\sin \left(x\right)\right]\cdot \cos \left(x\right)+\sin \left(x\right)\cdot \frac{d}{dx}\left[\cos \left(x\right)\right]$$

   Let $f=\sin \left(x\right)$ and $g=\cos \left(x\right)$. The product rule says $(f\cdot g{)}^{\prime }=f^{\prime }\cdot g+f\cdot g^{\prime }$ — find each factor's derivative, then assemble.

2. ↳Find f′: Differentiate the first factor, $f=\sin \left(x\right)$:
3. 2Derivative of sin
   $$\frac{d}{dx}\left[\sin \left(x\right)\right]=\cos \left(x\right)$$

   The derivative of $\sin (x)$ is $\cos (x)$ — a standard rule.

4. ↳Find g′: Differentiate the second factor, $g=\cos \left(x\right)$:
5. 3Derivative of cos
   $$\frac{d}{dx}\left[\cos \left(x\right)\right]=-\sin \left(x\right)$$

   The derivative of $\cos (x)$ is $-\sin (x)$ — a standard rule.

6. =Combine
   $$\frac{d}{dx}\left[\sin \left(x\right)\cdot \cos \left(x\right)\right]=\cos \left(x\right)\cdot \cos \left(x\right)+\sin \left(x\right)\cdot -\sin \left(x\right)={\cos \left(x\right)}^2-{\sin \left(x\right)}^2$$

   With $f^{\prime }=\cos \left(x\right)$ and $g^{\prime }=-\sin \left(x\right)$, substitute into $f^{\prime }\cdot g+f\cdot g^{\prime }$ and simplify.

7. ✓Final Answer
   $$\frac{d}{dx}\left[\sin \left(x\right)\cdot \cos \left(x\right)\right]={\cos \left(x\right)}^2-{\sin \left(x\right)}^2$$

   The first derivative simplifies to ${\cos \left(x\right)}^2-{\sin \left(x\right)}^2$.