A few days ago

# Derivative?

What is the derivative of sinx(sinx+cosx) ?

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A few days ago

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In other words, {d/dx} sinx(sinx + cosx) = ?

Recall: {d/dx} sinx = cosx, and {d/dx} cosx = -sinx:

{d/dx} sinx(sinx + cosx)

= {d/dx} sin^2(x) + sin(x)cos(x)

= {d/dx} sin^2(x) + {d/dx} sin(x)cos(x)

For the first term, use the chain rule, and for the second term, use the product rule to get the following:

= 2sin(x)cos(x) + [cos^2(x) + sin(x)(-sin(x))]

= 2sin(x)cos(x) – sin^2(x) + cos^2(x)

Now you can simplify…

Recall: cos^2(x) – sin^2(x) = cos(2x), and

Recall: 2sin(x)cos(x) = sin(2x), so:

2sin(x)cos(x) + cos^2(x) – sin^2(x)

= sin(2x) + cos(2x)

Therefore, {d/dx} sinx(sinx + cosx) = sin(2x) + cos(2x).

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