गुलीवर समुद्र की लहरों से संघर्ष करता हुआ कहाँ पहुँचा?
Answers
Since x(t) and y(t) are both differentiable functions of t, both limits inside the last radical exist. Therefore, this value is finite. This proves the chain rule at t=t0; the rest of the theorem follows from the assumption that all functions are differentiable over their entire domains.
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Closer examination of Equation 4.29 reveals an interesting pattern. The first term in the equation is
∂f
∂x
·
dx
dt
and the second term is
∂f
∂y
·
dy
dt
. Recall that when multiplying fractions, cancelation can be used. If we treat these derivatives as fractions, then each product “simplifies” to something resembling ∂f/dt. The variables xandy that disappear in this simplification are often called intermediate variables: they are independent variables for the function f, but are dependent variables for the variable t. Two terms appear on the right-hand side of the formula, and f is a function of two variables. This pattern works with functions of more than two variables as well, as we see later in this section.