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education/math/MATH1210 (calc 1)/Derivatives.md
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@ -152,24 +152,25 @@ This is used when you want to take the derivative of a function raised to a func
1. $\ln y = \ln((7x+2)^x)$ 1. $\ln y = \ln((7x+2)^x)$
2. $\ln y = x*\ln(7x + 2)$ 2. $\ln y = x*\ln(7x + 2)$
3. $\dfrac{dy}{dx} \dfrac{1}{y} = 3x*\dfrac{1}{7x + 2} * 3\ln(7x+2)$ 3. $\dfrac{dy}{dx} \dfrac{1}{y} = \dfrac{7x}{7x + 2} * \ln(7x+2)$
4. $\dfrac{dy}{dx} = (\dfrac{7x}{7x+2} * \ln(7x+2))\ln(7x+2)^x$
> Find the derivative of the function $y = (2x \sin x)^{3x}$ > Find the derivative of the function $y = (2x \sin x)^{3x}$
4. $\ln y = \ln (3x \sin x)^{3x}$ 5. $\ln y = \ln (3x \sin x)^{3x}$
5. $\ln y = 3x * \ln(2x \sin x)$* 6. $\ln y = 3x * \ln(2x \sin x)$*
6. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$ 7. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$
7. $\dfrac{1}{y} \dfrac{dy}{dx} = 3(\ln 2 + \ln x + \ln(\sin(x))) + 3x (0 + \dfrac{1}{x} + \dfrac{1}{\sin x} * \cos x)$j 8. $\dfrac{1}{y} \dfrac{dy}{dx} = 3(\ln 2 + \ln x + \ln(\sin(x))) + 3x (0 + \dfrac{1}{x} + \dfrac{1}{\sin x} * \cos x)$j
8. $\dfrac{dy}{dx} = (3\ln 2 + 3 \ln x + 3\ln \sin(x) + 3\ln(\sin(x) + 3x\cot(x))(2x\sin x)^{3x}$ 9. $\dfrac{dy}{dx} = (3\ln 2 + 3 \ln x + 3\ln \sin(x) + 3\ln(\sin(x) + 3x\cot(x))(2x\sin x)^{3x}$
# 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 $g(x) = x^2 + 3)$.
9. First find the derivative of the outside function function ($f(x) = x^4$): 10. First find the derivative of the outside function function ($f(x) = x^4$):
$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$ $$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$
10. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. 11. 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 $x^4$
@ -205,7 +206,7 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$
- 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:
11. Let $y$ be some function of $x$ 12. Let $y$ be some function of $x$
12. $\dfrac{d}{dx} x = 1$ 13. $\dfrac{d}{dx} x = 1$
13. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ 14. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\