The derivative of a composite function
Answers are for parents and teachers.
1
Find the derivative of y = (5 − 2x)⁸.
y′ = 8(5 − 2x)⁷ · (−2) = −16(5 − 2x)⁷.
2
Write the derivative of y = e^(x²) · sin 2x.
y′ = 2x e^(x²) sin 2x + 2e^(x²) cos 2x = 2e^(x²)(x sin 2x + cos 2x).
3
Compute the derivative of h(x) = (x³ − 2)⁴ at x = 1.
-12
4
Find the derivative of y = ln(x² + 1) and relate its sign to the sign of x.
y′ = 2x/(x² + 1). The denominator is always positive, so y′ has the same sign as x.
5
Find the derivative of y = tan 5x.
y′ = 5/cos²5x.