What is the derivative of $$y=9^{x^2-8}$$ with respect to x?

$$\frac{dy}{dx}=9^{x^2-8}\times\ln9\times2x$$ .

How do you find dy/dx for the equation $$x^3 + y^3 = 36$$ ?

Using implicit differentiation, you get $$\frac{dy}{dx} = -\frac{x^2}{y^2}$$ .

What is the derivative of $$y=(2^{3x+4})$$ ?

$$\frac{dy}{dx} = 3(\ln2)(2^{3x+4})$$ .

What is the derivative of $$xy + y^2 = 2$$ ?

Using implicit differentiation, you find $$\frac{dy}{dx} = -\frac{y}{x + 2y}$$ .

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 given by $$\frac{d}{dx}(uv) = u$$ .

What is the quotient rule in differentiation?

The quotient rule states that if you have a function that is the quotient of two functions u(x) and v(x), then the derivative is given by $$\frac{d}{dx}(\frac{u}{v}) = \frac{u$$ .

What is the chain rule in differentiation?

The chain rule states that if you have a composite function f(g(x)), then the derivative is given by $$\frac{d}{dx}f(g(x)) = f$$ .

What is the significance of $$\ln a$$ in differentiation?

$$\ln a$$ appears in the derivative of exponential functions, indicating the rate of growth of the function with respect to its base.

How do you differentiate an exponential function?

The derivative of an exponential function $$a^{f(x)}$$ is given by $$\frac{d}{dx}(a^{f(x)}) = a^{f(x)} \ln(a) f$$ .

What is the implicit function theorem?

The implicit function theorem provides conditions under which a relation defines a function implicitly, allowing for the differentiation of y with respect to x.

What is the relationship between implicit differentiation and inverse functions?

Implicit differentiation can be used to find the derivative of inverse functions by differentiating the equation that defines the inverse relationship.

12-14-23 Implicit Differentiation Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, without solving for y explicitly.

2.

FLASHCARD QUESTION

Front

How do you apply the chain rule in implicit differentiation?

Back

In implicit differentiation, when differentiating a term involving y, you multiply by dy/dx. For example, if you differentiate y^2, you get 2y(dy/dx).

3.

FLASHCARD QUESTION

Front

Back

4.

FLASHCARD QUESTION

Front

Back

5.

FLASHCARD QUESTION

Front

Back

6.

FLASHCARD QUESTION

Front

What is the formula for the tangent line at a point on a curve?

Back

7.

FLASHCARD QUESTION

Front

How do you find the slope of the tangent line using implicit differentiation?

Back

Differentiate both sides of the equation with respect to x, solve for dy/dx, and evaluate at the given point.

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