Calculus I · 2A · review

Implicit, Inverse, and Logarithmic Differentiation Review

Review implicit, inverse, and logarithmic differentiation review with mixed practice and links back to the exact lessons behind each skill.

Course progress44 of 49 core pages44 of 49

Section overview

Implicit, inverse, and logarithmic differentiation

What this section is building

Review implicit, inverse, and logarithmic differentiation review with mixed practice and links back to the exact lessons behind each skill.

Notice

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

Decide

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

Avoid

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

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

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Review

Summary

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

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Review

Related resources

Original instruction with traceable references.

Review

Related resources

Original instruction with traceable references.