Derivative of Composite Function/Examples/(3x+1)^2

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\paren {3 x + 1}^2} = 6 \paren {3 x + 1}$

Proof

Let $u = 3 x + 1$.

Let $y = u^2$.

Then we have:

$y = \paren {3 x + 1}^2$

and so:

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\)

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds 2 u \cdot 3\)

Derivative of Square Function, Derivative of Identity Function: Corollary

\(\ds \)

\(=\)

\(\ds 6 \paren {3 x + 1}\)

simplification

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chain rule (for differentiation)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chain rule (for differentiation)