For p(x) = m[n(x)] = m o n:

Find the derivative of (x^−1/2)³ using the Chain Rule.

Using Chain Rule with n(x) = x^−1/2 and m(x) = x³ it follows that

n'(x) = (−½)x^−3/2

and

m'[n(x)] = 3(x^−1/2)² = 3x⁻¹

p'(x) = m' [n(x)]n'(x) = 3x⁻¹(−½)x^−3/2

p'(x) = (−3/2)x^−5/2

Example #2:

Find the derivative of sin (1 + x²) using the Chain Rule.

Using Chain Rule with n(x) = 1 + x² and m(x) = sin x it follows that

n'(x) = 2x

and

m'[n(x)] = cos(1 + x²)

p'(x) = m' [n(x)]n'(x) = cos(1 + x²)2x

p'(x) = 2x cos(1 + x²)