Chain Rule

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), then the derivative y' is given by y' = f'(g(x)) * g'(x).

You use the Chain Rule when differentiating a function that is composed of another function, i.e., when there is a function within a function.

$$f'(x) = x^6 (5 + 8x)^2 (35 + 80x)$$

$$y' = 4x \text{cos}(2x^2)$$

$$y' = -5(6x + 5)(3x^2 + 5x)^4$$

If $$u = g(x)$$ , then the derivative is $$\frac{d}{dx}(\text{sin}(u)) = \text{cos}(u) \frac{du}{dx}$$ .

If $$u = g(x)$$ , then the derivative is $$\frac{d}{dx}(\text{cos}(u)) = -\text{sin}(u) \frac{du}{dx}$$ .

$$y' = \frac{g'(x)}{g(x)}$$ .

$$y' = \text{e}^{g(x)} g'(x)$$ .

If $$y = f(g(x))$$ , then $$y' = f'(g(x)) \times g'(x)$$ .