# Step-by-step Solution

## Find the derivative using the product rule $\frac{d}{dx}\left(5\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)\right)$

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### Videos

$20\sin\left(x\right)\cos\left(x\right)$

## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(5\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)\right)$
1

The sum of two terms multiplied by their difference is equal to the square of the first term minus the square of the second term. In other words: $(a+b)(a-b)=a^2-b^2$, where:

• The first term ($a$) is $\sin\left(x\right)$.
• The second term ($b$) is $\cos\left(x\right)$.
Then:

$\frac{d}{dx}\left(5\left(\sin\left(x\right)^2-\cos\left(x\right)^2\right)\right)$
2

Solve the product $5\left(\sin\left(x\right)^2-\cos\left(x\right)^2\right)$

$\frac{d}{dx}\left(5\sin\left(x\right)^2-5\cos\left(x\right)^2\right)$

$20\sin\left(x\right)\cos\left(x\right)$
$\frac{d}{dx}\left(5\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)\right)$

Product rule

~ 0.77 seconds