Implicit Differentiation ~ Calculus

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both dependent and independent variables.

We use implicit differentiation when it is difficult or impossible to solve for one variable in terms of another, allowing us to differentiate both sides of an equation with respect to the independent variable.

The derivative of $$y$$ with respect to $$x$$ is denoted as $$\frac{dy}{dx}$$ .

Differentiate both sides with respect to $$x$$ : \(2x + y + x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0\). Solve for $$\frac{dy}{dx}$$ .

The derivative of $$tan(x)$$ is $$sec^2(x)$$ .

The derivative of $$sec(x)$$ is $$sec(x)tan(x)$$ .

The derivative of $$cot(x)$$ is $$-csc^2(x)$$ .

The derivative of $$csc(x)$$ is $$-csc(x)cot(x)$$ .

The chain rule states that if a function $$y$$ is defined as $$y = f(g(x))$$ , then the derivative is $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$ .

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