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nacho-man
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Can someone please explain how the result is obtained from the first line
$r^2 = y^2 + x^2$
(refer to attached image)
View attachment 3404
$r^2 = y^2 + x^2$
(refer to attached image)
View attachment 3404
Can I confirm what you've done to get the $dr, dx, dy$ out like that?MarkFL said:Given:
\(\displaystyle r^2=x^2+y^2\)
And then differentiating in differential form, we find:
\(\displaystyle 2r\,dr=2x\,dx+2y\,dy\)
Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly defined. It is typically used when a function is defined implicitly by an equation, rather than being explicitly stated as a function of a single variable.
Implicit differentiation should be used when a function cannot be easily expressed as a single variable or when solving for a derivative of a function with multiple variables. It is also useful when dealing with equations that involve both dependent and independent variables.
Explicit differentiation involves finding the derivative of a function that is explicitly defined in terms of a single variable. Implicit differentiation, on the other hand, is used to find the derivative of a function that is not explicitly defined and involves multiple variables.
The first step in implicit differentiation is to differentiate both sides of the equation with respect to the independent variable. Then, use the chain rule and product rule to simplify the equation. Finally, solve for the derivative of the dependent variable by isolating it on one side of the equation.
Yes, implicit differentiation can be used for any type of function, as long as it is defined implicitly by an equation. It is a versatile technique that can be used to find the derivatives of polynomial, exponential, logarithmic, and trigonometric functions, among others.