Differentiate the equation $$-3\text{sin}(x) + 4\text{cos}(y) = 10$$ with respect to x.

$$\frac{dy}{dx} = -\frac{3\text{cos}(x)}{4\text{sin}(y)}$$ .

Find $$\frac{dy}{dx}$$ for the equation $$-7x + 9y = 12$$ .

$$\frac{dy}{dx} = \frac{7}{9}$$ .

Differentiate the equation $$4x + 12y = 10$$ with respect to x.

$$\frac{dy}{dx} = -\frac{1}{3}$$ .

What does $$\frac{dy}{dx}$$ represent?

$$\frac{dy}{dx}$$ represents the rate of change of y with respect to x, or the slope of the tangent line to the curve at a given point.

How do you solve for $$\frac{dy}{dx}$$ in implicit differentiation?

To solve for $$\frac{dy}{dx}$$ , isolate $$\frac{dy}{dx}$$ on one side of the equation after differentiating both sides, and then simplify.

What is the significance of the negative sign in $$\frac{dy}{dx} = -\frac{3}{1}$$ ?

The negative sign indicates that as x increases, y decreases, showing an inverse relationship between the two variables.

Differentiate the equation $$x^2 + y^2 = 25$$ with respect to x.

$$\frac{dy}{dx} = -\frac{x}{y}$$ .

What is the formula for the derivative of a product of two functions?

The product rule states that if u and v are functions of x, then $$\frac{d(uv)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$ .

What is the formula for the derivative of a quotient of two functions?

The quotient rule states that if u and v are functions of x, then $$\frac{d(\frac{u}{v})}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$ .

What is the importance of implicit differentiation in calculus?

Implicit differentiation is important because it allows us to find derivatives of functions that are not easily expressed in explicit form, enabling us to analyze relationships between variables in complex equations.

How do you apply the chain rule in implicit differentiation?

In implicit differentiation, when differentiating a term involving y, apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function, which is $$\frac{dy}{dx}$$ .

Implicit Differentiation Basic Practice 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 differentiate equations where the dependent and independent variables are not separated. It allows us to find the derivative of a function defined implicitly by an equation.

2.

FLASHCARD QUESTION

Front

How do you differentiate an equation implicitly?

Back

To differentiate an equation implicitly, apply the derivative to both sides of the equation with respect to the independent variable, treating the dependent variable as a function of the independent variable. Use the chain rule when differentiating terms involving the dependent variable.

3.

FLASHCARD QUESTION

Front

What is the chain rule in differentiation?

Back

4.

FLASHCARD QUESTION

Front

Back

5.

FLASHCARD QUESTION

Front

Back

6.

FLASHCARD QUESTION

Front

What is the derivative of a constant?

Back

The derivative of a constant is zero. This means that if a term does not change with respect to the variable, its derivative is zero.

7.

FLASHCARD QUESTION

Front

Back

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