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education/math/MATH1210 (calc 1)/Derivatives.md
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@ -28,7 +28,13 @@ The above formula can be used to find the *derivative*. This may also be referre
$$ y - y_1 = m(x-x_1) $$ $$ y - y_1 = m(x-x_1) $$
Given that $m = f'(a)$ and that $(x_1, y_1) = (a, f(a))$, you get the equation: Given that $m = f'(a)$ and that $(x_1, y_1) = (a, f(a))$, you get the equation:
$$ y - f(a) = f'(a)(x - a) $$ $$ y - f(a) = f'(a)(x - a) $$
As a more practical example, given an equation with a slope of $6$, As a more practical example, given an equation with a slope of $6$ at the point $(-2, -4)$:
$$ y - (-4) = 6(x - -2)$$
Solving for $y$ looks like this:
1. $y + 4 = 6(x + 2)$
2. $y = 6(x + 2) - 4$
3. $y = 6x + 12 - 4$
4. $y = 6x + 8$
# Line Types # Line Types
## Secant Line ## Secant Line
A **Secant Line** connects two points on a graph. A **Secant Line** connects two points on a graph.
@ -123,9 +129,9 @@ $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$
> Given the function $(x^2+3)^4$, find the derivative. > Given the function $(x^2+3)^4$, find the derivative.
Using the chain rule, the above function might be described as $f(g(x))$, where $f(x) = x^4$, and $g(x) = x^2 + 3)$. Using the chain rule, the above function might be described as $f(g(x))$, where $f(x) = x^4$, and $g(x) = x^2 + 3)$.
6. First find the derivative of the outside function function ($f(x) = x^4$): 5. First find the derivative of the outside function function ($f(x) = x^4$):
$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$ $$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$
7. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. 6. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$.
$$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$ $$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$
> Apply the chain rule to $x^4$ > Apply the chain rule to $x^4$
@ -161,8 +167,8 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$
- Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$. - Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$.
Given these facts: Given these facts:
8. Let $y$ be some function of $x$ 7. Let $y$ be some function of $x$
9. $\dfrac{d}{dx} x = 1$ 8. $\dfrac{d}{dx} x = 1$
10. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ 9. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\
## Examples ## Examples