3. Use the information from the previous problem to find the following derivatives: (a) ( frac{mathrm{d}}{mathrm{d} x}left[3 x^{2} an (x) ight] ) (b) ( frac{mathrm{d}}{mathrm{d} x}left[frac{x}{csc (x)} ight] ) 4. Use the product rule twice to show that if ( f, g, ) and ( h, ) are differentiable functions, then [ frac{mathrm{d}}{mathrm{d} x}[f(x) cdot g(x) cdot h(x)]=f^{prime}(x) cdot g(x) cdot h(x)+f(x) cdot g^{prime}(x) cdot h(x)+f(x) cdot g(x) cdot h^{prime}(x) ]
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Step 1: Find the derivative of \(3x^2 \tan(x) + 10x^2\). Show more…
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$\begin{array}{l}{\text { (a) Use the Product Rule twice to prove that if } f, g, \text { and } h \text { are }} \\ {\text { differentiable, then }(\mathrm{fgh})^{\prime}=\mathrm{f}^{\prime} g h+\mathrm{f} g^{\prime} h+\mathrm{f} g \mathrm{h}^{\prime} .}\end{array}$ $\begin{array}{r}{\text { (b) Taking } \mathrm{f}=g=\mathrm{h} \text { in part (a), show that }} \\ {\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{f}(\mathrm{x})]^{3}=3[\mathrm{f}(\mathrm{x})]^{2} \mathrm{f}^{\prime}(\mathrm{x})} \\ {\text { (c) Use part (b) to differentiate } \mathrm{y}=\mathrm{e}^{3 \mathrm{x}} \text { . }}\end{array}$
Differentiation Rules
The Product and Quotient Rules
(a) Use the Product Rule twice to prove that it $ f,g, $ and $ h $ are differentiable. then $ (fgh)' = f'gh + fgh'. + fgh'. $ (b) Taking $ f = g = h $ in part (a), show that $ \frac {d}{dx}[f(x)]^3=3[f(x)]^2f'(x) $ (c) Use part (b) to differentiate $ y = e^{3x}. $
Ahmed M.
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