⋔αth ∫∆rm

3.4

The Chain Rule

The Chain Rule

F'(x) = f'(g(x))×g'(x)

if g is differentiable at x and f is differentiable at g(x), then the composite function F = f * g defined by F(x) = f(g(x)) is differenciable at x and F' is given by the product above.

The Chain Rule (Leibniz notation)

dydx=dydududx

Same Chain Rule as above, but written in Leibniz notation. This is true if y = f(u) and u=g(x) are both differentiable functions.

The Chain Rule Combined With the Power Rule

ddx(un) = nun-1dudxor: ddx[g(x)]n=n[g(x)]n-1×g'(x)

if n is and real number and u = g(x) is differentiable. This also means that ddx(ax) = axln a

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