Product and Quotient Rules (Derivatives)

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

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: \(\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).

Using the Product Rule, the derivative is: $$8x^3 - 3x^2 - 6x$$ .

Using the Quotient Rule, the derivative is: $$\frac{x\cos x - 2\sin x}{x^3}$$ .

To find the slope of the tangent line at a point (a, f(a)), compute the derivative f'(x) and evaluate it at x = a.

The derivative is: $$\frac{df(x)}{dx} = 6x + 4\sec^2 x$$ .

The derivative represents the rate of change of a function with respect to its variable, indicating how the function behaves locally.

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point.

Using the Product Rule, the derivative is: $$-21x^6 + 6x^2$$ .

The Chain Rule states that if a function y = f(g(x)) is composed of two functions, then its derivative is: \(\frac{dy}{dx} = f'(g(x))g'(x)\).