No formula for y? Differentiate anyway.

Some curves can't be written as $y = \dots$ at all. They still have a tangent at every point, and the chain rule finds its gradient without ever solving for $y$.

scroll ↓

$x^2 + y^2 = 25$ is a circle of radius $5$. It isn't $y = \dots$: most values of $x$ give two values of $y$. Yet at every point there is a tangent with a gradient.

Differentiate each term with respect to $x$. The chain rule turns $y^2$ into $2y\dfrac{dy}{dx}$, because $y$ depends on $x$:

As the point moves round, the gradient $-\tfrac{x}{y}$ gets steeper. At $(5, 0)$ the formula would divide by $0$: the tangent is vertical and $\tfrac{dy}{dx}$ does not exist. At the top, $(0, 5)$, it is $0$.

Terms that mix $x$ and $y$ need the product rule too. For $x^3 + xy + y^3 = 11$ (Example 13.4),

which is $-\tfrac{5}{13}$ at $(1, 2)$.

Every y term picks up a dy/dx.

Parametric differentiation, connected rates of change, concave and convex curves, points of inflection, and twelve graded exam questions with full worked solutions are waiting.

Read the full chapter › Jump to the questions ›