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Differentiate the following functions using product rule.​

Differentiate the following functions using product rule.​-example-1
User Jenea Vranceanu
by
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1 Answer

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13 votes

d) By the product rule,


y = 5 e^x \sin(x) \implies (dy)/(dx) = (d5)/(dx) e^x \sin(x) + 5 (de^x)/(dx) \sin(x) + 5 e^x (d\sin(x))/(dx)

The derivative of a constant is 0; the derivative of
e^x is
e^x; the derivative of
\sin(x) is
\cos(x). So


(dy)/(dx) = \boxed{5 e^x \sin(x) + 5 e^x \cos(x)}

e) Rewrite
\sqrt x = x^(1/2). By the product rule,


y = \sqrt x\,e^x = x^(1/2) e^x \implies (dy)/(dx) = (dx^(1/2))/(dx) e^x + x^(1/2) (de^x)/(dx)

By the power rule,


(dx^(1/2))/(dx) = \frac12 x^(1/2-1) = \frac12 x^(-1/2)

Then


(dy)/(dx) = \boxed{(e^x)/(2\sqrt x) + \sqrt x\,e^x}

f) By the product rule,


y = (1 + \sin(x)) \tan(x) \implies (dy)/(dx) = (d(1+\sin(x)))/(dx) \tan(x) + (1+\sin(x)) (d\tan(x))/(dx)

The derivative of
\tan(x) is
\sec^2(x). So


(dy)/(dx) = \cos(x)\tan(x) + (1 + \sin(x)) \sec^2(x)

which we can simplify using


\tan(x) = (\sin(x))/(\cos(x))

and expand to get


(dy)/(dx) = \boxed{\sin(x) + \sec^2(x) + \tan(x) \sec(x)}

User Chris Hamilton
by
3.2k points
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