Taking second derivatives in Calc 1 wasn't so bad. Here it can be rather tedious and challenging depending on the number of variables and parameters. To take the simplest case, suppose we have an equation in two variables f(x, y) = 0, which defines y as an implicit function of x, twice differentiable. (a) Starting with the first derivative, d²y_2fxfyfry - ffxx - fzfyy f3 dx² dy dx = fx fy where fr = apply the quotient rule and chain rule to show: af მე 2 af = მყ " fxy = Ꭷ f მყმუ and so on. (b) Use the above formula (whether or not you figured it out) to find d²y for the equation x²+y² = 4, dx² even though you probably did this in Calc 1 without such a monstrous formula.
Taking second derivatives in Calc 1 wasn't so bad. Here it can be rather tedious and challenging depending on the number of variables and parameters. To take the simplest case, suppose we have an equation in two variables f(x, y) = 0, which defines y as an implicit function of x, twice differentiable. (a) Starting with the first derivative, d²y_2fxfyfry - ffxx - fzfyy f3 dx² dy dx = fx fy where fr = apply the quotient rule and chain rule to show: af მე 2 af = მყ " fxy = Ꭷ f მყმუ and so on. (b) Use the above formula (whether or not you figured it out) to find d²y for the equation x²+y² = 4, dx² even though you probably did this in Calc 1 without such a monstrous formula.
Taking second derivatives in Calc 1 wasn't so bad. Here it can be rather tedious and challenging depending on the number of variables and parameters. To take the simplest case, suppose we have an equation in two variables f(x, y) = 0, which defines y as an implicit function of x, twice differentiable. (a) Starting with the first derivative, d²y_2fxfyfry - ffxx - fzfyy f3 dx² dy dx = fx fy where fr = apply the quotient rule and chain rule to show: af მე 2 af = მყ " fxy = Ꭷ f მყმუ and so on. (b) Use the above formula (whether or not you figured it out) to find d²y for the equation x²+y² = 4, dx² even though you probably did this in Calc 1 without such a monstrous formula.
Transcribed Image Text:Taking second derivatives in Calc 1 wasn't so bad. Here it can be rather tedious and challenging
depending on the number of variables and parameters. To take the simplest case, suppose we have an
equation in two variables f(x, y) = 0, which defines y as an implicit function of x, twice differentiable.
(a) Starting with the first derivative,
d²y_2fxfyfry - ffxx - fzfyy
f3
dx²
dy
dx
=
fx
fy
where fr
=
apply the quotient rule and chain rule to show:
af
მე
2
af
=
მყ
"
fxy
=
Ꭷ f
მყმუ
and so on.
Transcribed Image Text:(b) Use the above formula (whether or not you figured it out) to find
d²y
for the equation x²+y² = 4,
dx²
even though you probably did this in Calc 1 without such a monstrous formula.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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