How do you differentiate  y = (6e^(-7x)+2x)^2 using the chain rule?

Sep 15, 2016

Answer:

$y ' = - 504 {e}^{- 14 x} + 12 {e}^{- 7 x} - 84 x {e}^{- 7 x} + 4 x$

Explanation:

To differentiate the given function $y$ using chain rule let:
$f \left(x\right) = {x}^{2}$ and
$g \left(x\right) = 6 {e}^{- 7 x} + 2 x$
So, $y = f \left(g \left(x\right)\right)$

To differentiate $y = f \left(g \left(x\right)\right)$ we have to use chain rule as follows:
Then $y ' = \left(f \left(g \left(x\right)\right)\right) ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Let's find $f ' \left(x\right)$ and $g ' \left(x\right)$
$f ' \left(x\right) = 2 x$
$g ' \left(x\right) = - 7 \cdot 6 {e}^{- 7 x} + 2 = - 42 {e}^{- 7 x} + 2$

$y ' = \left(f \left(g \left(x\right)\right)\right) ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

$y ' = 2 \left(6 {e}^{- 7 x} + 2 x\right) \cdot \left(- 42 {e}^{- 7 x} + 2\right)$
$y ' = 2 \left(- 252 {e}^{- 14 x} + 12 {e}^{- 7 x} - 84 x {e}^{- 7 x} + 4 x\right)$
$y ' = - 504 {e}^{- 14 x} + 12 {e}^{- 7 x} - 84 x {e}^{- 7 x} + 4 x$