Home page for accesible maths 3 Chapter 3 contents 3.23 Gradient of the tangent to a curve 3.25 Examples of implicit differentiation

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3.24 Proof of the gradient formula

To find how $y$ varies with $x$ , we set $x=t$ and $y=y(t)$ so that $(t,y(t))$ describes $C$ near to $(a,b)$ as $t$ varies. Then $0=f(t,y(t))$ for all $t$ near $a$ and it follows from the chain rule that

0={{\partial f}\over{\partial x}}{{dx}\over{dt}}+{{\partial f}\over{\partial y% }}{{dy}\over{dt}};

so that the derivative of the implicitly defined function is

{{dy}\over{dx}}={{dy/dt}\over{dx/dt}}=-{{f_{x}}\over{f_{y}}},

which equals the gradient of the tangent to the curve $C$ .

To differentiate implicitly means to write down

f_{x}+f_{y}{{dy}\over{dx}}=0,

without attempting to find an explicit formula for $y$ in terms of $x$ .