Derivatives Review

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. It represents the slope of the tangent line to the function's graph at that point.

The power rule states that if $$f(x) = x^n$$ , then the derivative $$f'(x) = n \times x^{n-1}$$ .

The product rule states that if $$f(x) = g(x) \times h(x)$$ , then the derivative is $$f'(x) = g'(x) \times h(x) + g(x) \times h'(x)$$ .

The chain rule states that if $$f(g(x))$$ is a composite function, then the derivative is $$f'(g(x)) \times g'(x)$$ .

Using the product rule and chain rule, $$f'(x) = (x^2 - 2)^3 + (6x^2 + 6x)(x^2 - 2)^2$$ .

The quotient rule states that if $$f(x) = \frac{g(x)}{h(x)}$$ , then the derivative is $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$ .

A function is differentiable at a point if it has a derivative at that point, meaning the function is smooth and has no sharp corners or discontinuities there.

The first derivative test is used to determine the local maxima and minima of a function by analyzing the sign of the derivative before and after critical points.

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, allowing us to find $$\frac{dy}{dx}$$ without solving for y explicitly.

The second derivative is the derivative of the derivative of a function, denoted as $$f''(x)$$ , and it provides information about the concavity of the function.