Implicit Differentiation Practice

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly as y = f(x). It involves taking the derivative of both sides of the equation and applying the chain rule.

The chain rule states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x: $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$ .

To find $$\frac{dy}{dx}$$ , differentiate both sides: 2y $$\frac{dy}{dx} = 10$$ . Thus, $$\frac{dy}{dx} = \frac{10}{2y} = \frac{5}{y}$$ .

Using the chain rule, $$y' = 2 \times 2x \times \text{cos}(2x^2) = 4x \text{cos}(2x^2)$$ .

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

Finding $$\frac{dy}{dx}$$ at a specific point gives the slope of the tangent line to the curve at that point, indicating the rate of change of y with respect to x.

The product rule states that if $$u(x)$$ and $$v(x)$$ are functions, then the derivative of their product is: $$\frac{d}{dx}(uv) = u'v + uv'$$ .

The quotient rule states that if $$u(x)$$ and $$v(x)$$ are functions, then the derivative of their quotient is: $$\frac{d}{dx}\frac{u}{v} = \frac{u'v - uv'}{v^2}$$ .

The derivative of $$e^{f(x)}$$ is given by the chain rule: $$\frac{d}{dx} e^{f(x)} = e^{f(x)} f'(x)$$ .

The derivative of $$\text{ln}(f(x))$$ is given by: $$\frac{d}{dx} \text{ln}(f(x)) = \frac{f'(x)}{f(x)}$$ .