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education/math/MATH1210 (calc 1)/Derivatives.md
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@ -122,6 +122,8 @@ $$ \dfrac{d}{dx}(\dfrac{f(x)}{g(x)}) = \dfrac{f'(x)g(x) -f(x)g'(x)}{(g(x))^2} $
$$ \dfrac{d}{dx} e^x = e^x $$ $$ \dfrac{d}{dx} e^x = e^x $$
$$ \dfrac{d}{dx}a^x = a^x*(\ln(a)) $$ $$ \dfrac{d}{dx}a^x = a^x*(\ln(a)) $$
for all $a > 0$ for all $a > 0$
# Logarithms # Logarithms
For natural logarithms: For natural logarithms:
@ -129,17 +131,27 @@ $$ \dfrac{d}{dx} \ln |x| = \dfrac{1}{x} $$
For other logarithms: For other logarithms:
$$ \dfrac{d}{dx} \log_a x = \dfrac{1}{(\ln a) x}$$ $$ \dfrac{d}{dx} \log_a x = \dfrac{1}{(\ln a) x}$$
When solving problems that make use of logarithms, consider making use of logarithmic properties to make life easier, eg: When solving problems that make use of logarithms, consider making use of logarithmic properties to make life easier:
$$ \ln(\dfrac{x}{y}) = \ln(x) - \ln(y) $$ $$ \ln(\dfrac{x}{y}) = \ln(x) - \ln(y) $$
$$ \ln(a^b) = b\ln(a) $$
## Logarithmic Differentiation
This is used when you want to take the derivative of a function raised to a function ($f(x)^{g(x)})$
1. $\dfrac{d}{dx} x^x$
2. $y = x^x$
3. $\ln y = \ln x^x$
4. $\ln(y) = x*\ln(x)$
5. $\dfrac{d}{dx} \ln y = \dfrac{d}{dx} x \ln x$
6. $\dfrac{1}{y} \dfrac{dy}{dx} = 1 * \ln x + x * \dfrac{1}{x}$
# Chain Rule # Chain Rule
$$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$ $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$
## Examples ## Examples
> Given the function $(x^2+3)^4$, find the derivative. > Given the function $(x^2+3)^4$, find the derivative.
Using the chain rule, the above function might be described as $f(g(x))$, where $f(x) = x^4$, and $g(x) = x^2 + 3)$. Using the chain rule, the above function might be described as $f(g(x))$, where $f(x) = x^4$, and $g(x) = x^2 + 3)$.
5. First find the derivative of the outside function function ($f(x) = x^4$): 7. First find the derivative of the outside function function ($f(x) = x^4$):
$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$ $$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$
6. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. 8. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$.
$$ \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)$$
> Apply the chain rule to $x^4$ > Apply the chain rule to $x^4$
@ -175,7 +187,7 @@ $$ \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:
7. Let $y$ be some function of $x$ 9. Let $y$ be some function of $x$
8. $\dfrac{d}{dx} x = 1$ 10. $\dfrac{d}{dx} x = 1$
9. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ 11. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\