What is the derivative of $$y$$ with respect to $$x$$ in implicit differentiation?

The derivative of $$y$$ with respect to $$x$$ is denoted as $$\frac{dy}{dx}$$ .

How do you differentiate $$x^2 + xy + y^3 = 0$$ using implicit differentiation?

Differentiate both sides with respect to $$x$$ : $2x + y + x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$. Solve for $$\frac{dy}{dx}$$ .

What is the derivative of $$tan(x)$$ ?

The derivative of $$tan(x)$$ is $$sec^2(x)$$ .

What is the derivative of $$sec(x)$$ ?

The derivative of $$sec(x)$$ is $$sec(x)tan(x)$$ .

What is the derivative of $$cot(x)$$ ?

The derivative of $$cot(x)$$ is $$-csc^2(x)$$ .

What is the derivative of $$csc(x)$$ ?

The derivative of $$csc(x)$$ is $$-csc(x)cot(x)$$ .

What is the chain rule in differentiation?

The chain rule states that if a function $$y$$ is defined as $$y = f(g(x))$$ , then the derivative is $$\frac{dy}{dx} = f$$ .

What is the product rule in differentiation?

The product rule states that if you have two functions $$u(x)$$ and $$v(x)$$ , then the derivative of their product is $$\frac{d}{dx}[u(x)v(x)] = u$$ .

What is the quotient rule in differentiation?

The quotient rule states that if you have two functions $$u(x)$$ and $$v(x)$$ , then the derivative of their quotient is $$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u$$ .

What is the significance of the derivative in calculus?

The derivative represents the rate of change of a function with respect to its variable, indicating how a function behaves locally.

What is an implicit function?

An implicit function is a function defined by an equation that relates the dependent and independent variables without explicitly solving for one variable in terms of the other.

What is the relationship between implicit differentiation and the slope of a curve?

Implicit differentiation allows us to find the slope of a curve defined by an implicit equation at any point by calculating $$\frac{dy}{dx}$$ .

How do you apply implicit differentiation to find critical points?

To find critical points, use implicit differentiation to find $$\frac{dy}{dx}$$ , set it equal to zero, and solve for $$x$$ and $$y$$ .

What is the importance of understanding derivatives in real-world applications?

Understanding derivatives is crucial for analyzing rates of change in various fields such as physics, engineering, and economics.

Implicit Differentiation ~ Calculus Flashcards

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1.

FLASHCARD QUESTION

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What is Implicit Differentiation?

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Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both dependent and independent variables.

2.

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When do we use Implicit Differentiation?

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We use implicit differentiation when it is difficult or impossible to solve for one variable in terms of another, allowing us to differentiate both sides of an equation with respect to the independent variable.

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