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education/math/MATH1210 (calc 1)/Derivatives.md
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@ -158,9 +158,9 @@ This is used when you want to take the derivative of a function raised to a func
> Find the derivative of the function $y = (2x \sin x)^{3x}$ > Find the derivative of the function $y = (2x \sin x)^{3x}$
5. $\ln y = \ln (3x \sin x)^{3x}$ 5. $\ln y = \ln (3x \sin x)^{3x}$
6. $\ln y = 3x * \ln(2x \sin x)$* 6. $\ln y = 3x * \ln(2x \sin x)$
7. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$ 7. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$
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. $\dfrac{1}{y} \dfrac{dy}{dx} = 3(\ln 2 + \ln x + \ln(\sin(x))) + 3x (0 + \dfrac{1}{x} + \dfrac{1}{\sin x} * \cos x)$
9. $\dfrac{dy}{dx} = (3\ln 2 + 3 \ln x + 3\ln \sin(x) + 3\ln(\sin(x) + 3x\cot(x))(2x\sin x)^{3x}$ 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
$$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$ $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$
@ -209,5 +209,5 @@ $$ \dfrac{d}{dx}(\arcsin(x) = \dfrac{1}{\sqrt{1-x^2}}$$
Given these facts: Given these facts:
12. Let $y$ be some function of $x$ 12. Let $y$ be some function of $x$
13. $\dfrac{d}{dx} x = 1$ 13. $\dfrac{d}{dx} x = 1$
14. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ 14. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$