Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 3

Express the composite function $y = (1-x^2)^{10}$ in the form $f(g(x))$. [Identify the inner function $u=g(x)$ and the outer function $y = f(u)$.] Then find the derivative $\displaystyle \frac{dy}{dx}$
Let $y = f(g(x))$ where $u = g(x) = 1 - x^2$ and $y = f(u) = u^{10}$

Then,

$
\begin{equation}
\begin{aligned}
y' &= \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}\\
\\
y' &= \frac{d}{du}(u^{10}) \cdot \frac{d}{dx}(1-x^2)\\
\\
y' &= (10u^{9})(-2x) && \text{Simplify the equation}\\
\\
y' &= -20x u^{9} && \text{Substitute value of } u\\
\\
y' &= -20x ( 1-x^2)^{9}
\end{aligned}
\end{equation}
$

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