dy dx= dy du du dx Differentiate the outer function, then multiply by the derivative of the inner function. Example Differentiate y=(x2+1)10 with respect to x. Let u=x2+1, so y=u10. dudx=2x dydu=10u9 dydx = dydu dudx =10u9(2x) =20x(x2+1)9 For the smaller power y=(x2+1)2, expansion is another way to differentiate. y =x4+2x2+1 dydx=4x3+4x

\(\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm du}\dfrac{\mathrm du}{\mathrm dx}\)

Differentiate the outer function, then multiply by the derivative of the inner function.

Differentiate \(y=(x^2+1)^{10}\) with respect to \(x\).

\(\dfrac{\mathrm{d}u}{\mathrm{d}x}=2x\qquad\dfrac{\mathrm{d}y}{\mathrm{d}u}=10u^9\)

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)\({}=\dfrac{\mathrm{d}y}{\mathrm{d}u}\dfrac{\mathrm{d}u}{\mathrm{d}x}\)\({}=10u^9(2x)\)\({}=20x(x^2+1)^9\)

For the smaller power \(y=(x^2+1)^2\), expansion is another way to differentiate.

\(y\)\({}=x^4+2x^2+1\)

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=4x^3+4x\)

Chain Rule

Chain Rule

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