Mathematics

The constant factors out: d/dx[c·f(x)] = c·f'(x). Constants ride along through differentiation.

d/dx[xⁿ] = n·x^(n−1). Multiply by the exponent, then decrease the exponent by one.

d/dx[u·v] = u'v + uv'. Differentiate each factor in turn while holding the other fixed, then add.

d/dx[u/v] = (u'v − uv') / v². Remember 'low d-high minus high d-low, over low squared.'

d/dx[f(g(x))] = f'(g(x))·g'(x). Differentiate the outer function, then multiply by the derivative of the inner function.

d/dx[sin x] = cos x and d/dx[cos x] = −sin x. The cosine derivative carries a negative sign.