Quick questions on Differentiating an inverse function: the rule dy/dx = 1/(dx/dy), gradients of an inverse at a point, and the second derivative of an inverse

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is chain rule on

f ) = x

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By definition, applying

f

f^{-1}(x)

returns

x

. Differentiate both sides with respect to

x

, using the chain rule on the left:

What is the

dx/dy

form?

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Write

y = f^{-1}(x)

, which is the same as

x = f(y)

. Differentiating

x = f(y)

with respect to

y

gives

\frac{dx}{dy} = f'(y)

, and treating the derivative as a ratio,

What is reflection intuition?

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The graph of

f^{-1}

is the graph of

f

reflected in the line

y = x

. Reflection in

y = x

swaps the horizontal and vertical directions, so it swaps the rise and the run of any tangent line. A tangent of gradient

\frac{\text{rise}}{\text{run}}

f

becomes a tangent of gradient

\frac{\text{run}}{\text{rise}}

f^{-1}

, that is, the reciprocal.

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