Missing dy/dx on y-terms
\(\frac{d}{dx}(y^3)=3y^2\frac{dy}{dx}\), and \(\frac{d}{dx}(xy)=y+x\frac{dy}{dx}\).
Continuity and Differentiability
Derivatives of Implicit Functions
Understand
Sometimes \(y\) is not given explicitly, as in \(x^2+y^2=25\) or \(x^3+y^3=6xy\). Treat \(y\) as a function of \(x\) and differentiate every term with respect to \(x\), using the chain rule on terms containing \(y\): \(\frac{d}{dx}(y^2)=2y\frac{dy}{dx}\), \(\frac{d}{dx}(xy)=x\frac{dy}{dx}+y\). Then collect the \(\frac{dy}{dx}\) terms and solve.
Example: \(x^2+y^2=25\Rightarrow2x+2y\,y\,'=0\Rightarrow y\,'=-\frac{x}{y}\). At \((3,4)\) the slope is \(-\frac34\).
Key Concepts
- Every y-term picks up a factor dy/dx.
- Use the product rule for xy, x²y, etc.
- Solve the resulting linear equation for dy/dx.
Common Mistakes
Practice & Topic Test
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Differentiate both sides term by term, remembering the chain rule for y, then solve for dy/dx.
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