Math · Calculus I · Worked example
Find dy/dx on a circle and its tangent line
For the circle x² + y² = 25, find dy/dx by implicit differentiation, then the tangent line at (3, 4).
Check that the point is on the curve
3² + 4² = 9 + 16 = 25, so (3, 4) lies on the circle.
Differentiate both sides
The derivative of x² is 2x. The derivative of y² is 2y·dy/dx by the chain rule. The constant 25 has derivative 0.
Solve for dy/dx
Subtract 2x from both sides and divide by 2y, which requires y ≠ 0.
Evaluate at (3, 4)
Substitute x = 3 and y = 4: the slope of the tangent is −3/4.
Write the tangent line
Use point-slope form through (3, 4) with slope −3/4, then expand.
Check with geometry
The radius to (3, 4) has slope 4/3. A tangent to a circle is perpendicular to the radius, and (4/3)(−3/4) = −1, so the slopes agree with the geometry.
Result
dy/dx = −x/y, and the tangent line at (3, 4) is y = −(3/4)x + 25/4.
Your turn
Find dy/dx for x² + xy + y² = 7, and the slope at (1, 2).
Show the answer and explanation
dy/dx = −(2x + y)/(x + 2y); the slope at (1, 2) is −4/5.
Differentiate: 2x + (y + x·dy/dx) + 2y·dy/dx = 0. Collect: (x + 2y)dy/dx = −(2x + y), so dy/dx = −(2x + y)/(x + 2y). At (1, 2), which is on the curve since 1 + 2 + 4 = 7, the slope is −4/5.
Keep exploring
Plot the circle in Implicit curves, then move the point: wherever y is small the slope −x/y grows large, and at (5, 0) the tangent is vertical.
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Plot the circle in Implicit curves Graph the circle and its tangent Write the steps in Math Open worked example on a board Implicit differentiation in Math ReferenceYour existing work stays on this device. Examples open as editable copies.