# How do you differentiate f(x)=1/e^sqrt(1-(3x+5)^2) using the chain rule.?

$\textcolor{b l u e}{f ' \left(x\right) = \frac{- \left(3 x + 5\right) \cdot \sqrt{1 - {\left(3 x + 5\right)}^{2}} \cdot {e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}}}{3 {x}^{2} + 10 x + 8}}$

#### Explanation:

The given equation is $f \left(x\right) = \frac{1}{e} ^ \left(\sqrt{1 - {\left(3 x + 5\right)}^{2}}\right)$

The solution:

$f \left(x\right) = \frac{1}{e} ^ \left(\sqrt{1 - {\left(3 x + 5\right)}^{2}}\right)$

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\frac{1}{e} ^ \left(\sqrt{1 - {\left(3 x + 5\right)}^{2}}\right)\right) = \frac{d}{\mathrm{dx}} \left({e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}}\right)$

$f ' \left(x\right) = {e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}} \cdot \frac{d}{\mathrm{dx}} \left(- \sqrt{1 - {\left(3 x + 5\right)}^{2}}\right)$

$f ' \left(x\right) = {e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}} \cdot \left(- 1\right) \cdot \frac{d}{\mathrm{dx}} \left(\sqrt{1 - {\left(3 x + 5\right)}^{2}}\right)$

$f ' \left(x\right) =$

${e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}} \cdot \left(- \frac{1}{2 \sqrt{1 - {\left(3 x + 5\right)}^{2}}}\right) \frac{d}{\mathrm{dx}} \left(1 - {\left(3 x + 5\right)}^{2}\right)$

$f ' \left(x\right) =$

${e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}} \cdot \left(- \frac{1}{2 \sqrt{1 - {\left(3 x + 5\right)}^{2}}}\right) \left(0 - 2 {\left(3 x + 5\right)}^{1} \cdot 3\right)$

$f ' \left(x\right) =$

${e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}} \cdot \left(- \frac{1}{2 \sqrt{1 - {\left(3 x + 5\right)}^{2}}}\right) \left(- 6 \left(3 x + 5\right)\right)$

$f ' \left(x\right) = {e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}} \cdot \left(\frac{9 x + 15}{\sqrt{1 - {\left(3 x + 5\right)}^{2}}}\right)$

We simplify by rationalization

$f ' \left(x\right) = \frac{- \left(9 x + 15\right) \cdot \sqrt{1 - {\left(3 x + 5\right)}^{2}} \cdot {e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}}}{9 {x}^{2} + 30 x + 24}$

Reducing the numbers by canceling common numerical coefficients

$\textcolor{b l u e}{f ' \left(x\right) = \frac{- \left(3 x + 5\right) \cdot \sqrt{1 - {\left(3 x + 5\right)}^{2}} \cdot {e}^{- \sqrt{1 - {\left(3 x + 5\right)}^{2}}}}{3 {x}^{2} + 10 x + 8}}$

God bless....I hope the explanation is useful.