Quick questions on Further differentiation and applications: QCE Maths Methods Unit 4

4short 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 are implicit derivative for shape constraints?

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If a curve is defined implicitly by

x^2 + y^2 + xy = 10

, find

dy/dx

at the point

(1, 2)

(which satisfies the equation:

1 + 4 + 2 = 7

- let me re-check:

(1)^2 + (2)^2 + (1)(2) = 1 + 4 + 2 = 7 \neq 10

. Use

(2, 1)

4 + 1 + 2 = 7

, still not. Skip the worked specific point; use general approach).

What is logarithmic differentiation without final

y

substitution?

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After computing

\frac{1}{y} \frac{dy}{dx}

, you must multiply by

y

and replace

y

with the original expression.

What is power rule applied to variable exponent?

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\frac{d}{dx}(x^x)

is NOT

x \cdot x^{x-1} = x^x

. The power rule applies only to constant exponents. Use logarithmic differentiation.

What is sign error in inverse function rule?

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The derivative of

f^{-1}

1 / f'(y)

, not

1 / f'(x)

. The

y

is the value of the inverse function at

x

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