Implicit Differentiation Basic Practice

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.

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.

The chain rule states that if a function y is defined as a function of u, which is in turn a function of x, then the derivative of y with respect to x is given by $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$ .

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

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

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.

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

$$\frac{dy}{dx}$$ 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.

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

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