Implicit Differentiation ~ Calculus

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, rather than explicitly solving for y.

To apply implicit differentiation, differentiate both sides of the equation with respect to x, treating y as a function of x. Use the chain rule for terms involving y.

$$y' = -\frac{2x + y}{x + 3y^2}$$ .

$$sec(x)tan(x)$$ .

$$sec^2(x)$$ .

The function is decreasing.

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: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ .

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

The quotient rule states that if you have a function that is the quotient of two functions u(x) and v(x), the derivative is given by: $$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$$ .

A function is increasing on an interval if its derivative is positive throughout that interval.