Calculus I · Unit 2A · review

Implicit, Inverse, and Logarithmic Differentiation Review

Review

Summary

• Implicit differentiation treats $y$ as a function of $x$ , so every derivative of a $y$ -expression includes $y'$ . • Inverse-function slopes are reciprocal at corresponding points. • Inverse trig derivatives follow from implicit differentiation and identities. • Logarithmic differentiation converts products to sums, quotients to differences, and powers to coefficients.

Exercise

Find $y'$ for $x^3+y^3=6xy$ .

Exercise

Find the tangent line to $x^2+xy+y^2=7$ at $(1,2)$ .

Exercise

Find $y"$ for $x^2+y^2=9$ .

Exercise

If $f(3)=8$ and $f'(3)=-2$ , find $(f^{-1})'(8)$ .

Exercise

Differentiate $\arcsin(2x)$ .

Exercise

Differentiate $\arctan(x^2)$ .

Exercise

Use logarithmic differentiation on $y=x^2(x+1)^3/(x^2+4)$ .

Exercise

Differentiate $(x+2)^{x^2}$ .

After the explanation

Use the section idea

Reading lens

Track which variable depends on which and use reciprocal or logarithmic structure only where its conditions hold.

Mental model

Implicit equations constrain variables together; inverse functions exchange inputs and outputs; logarithms turn products and powers into sums.

Decision

Choose implicit, inverse, or logarithmic differentiation from the equation's representation, not from surface complexity.

Common trap

Dropping a y-prime factor, using a reciprocal slope at the wrong point, or ignoring domain restrictions.

Check yourself

Can you identify the correspondence point and all hidden dependencies before differentiating?

Source & rights

Original instruction with traceable references.

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