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# If $f$ is a differentiable function, find an expression for the derivative of each of the following functions.(a) $y = x^2 f(x)$(b) $y = \frac {f(x)}{x^2}$(c) $y = \frac {x^2}{f(x)}$(d) $y = \frac {1+2 xf(x)}{\sqrt{x}}$

## (a) $\frac{d y}{d x}=x^{2} \cdot f^{\prime}(x)+2 x \cdot f(x)$(b) $\frac{f^{\prime}(x) x-2 f(x)}{x^{3}}$(c) $y^{\prime}=\frac{2 x f(x)-x^{2} f^{\prime}(x)}{(f(x))^{2}}$(d) $\frac{2 x^{2} f^{\prime}(x)+x f(x)-1}{2 x^{3 / 2}}$

Derivatives

Differentiation

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### Video Transcript

Hey, it's clear. So when you read here So for a we have why is equal to X square fffx we're gonna different She both sides in respect to x and we're gonna use the product rule So we get X square turns the over D X stuff of X plus f of X de over de X for X square This is equal to X square turns the derivative of F left to x times ffx part B we have why equals that the X divided by x square We're gonna use the quotient rule with you being equal to half of X, that makes the derivative equal to the derivative. Uh huh V is equal to x square, so that makes the derivative to X. So the derivative of Y is equal to the derivative of you. Tom's B minus you turns the derivative of the all over the square and this becomes equal to the derivative of ah turns X square minus F of x turns to x over next square square, which is equal to ah, the derivative turn sex minus two f of X over X cubed her part. See, we have the quotient role So we're gonna apply the quotient rule We're gonna make why Equal to u v So that means the derivative of y is equal to we turned the dirt of you Minus you terms the derivative of V over the square. So the derivative of why we get enough of us turns the derivative of X square minus X squared times the derivative of over of x square When we simplifying to get why is equal to two x fffx minus X square owns the derivative over of X square. For a party, we have the quotient role. So let's say we could divide first. Why is equal to one plus X f of X over a square of X which is equal to X to the negative 1/2 list X to the half terms Fffx. So we're gonna use the power on you Get the door if it ISS which is equal to negative 1/2 X. It's the negative three house plus next door in half Derivative Fffx plus next to the one, Huh? Thanks to drive it is of of X and then we can make this equal to No, I gotta warm over to x three House plus f of X over too next, the one her terms that's over X plus X. Still In her comes the derivative of arms to X to the three halves. Divide it by two x to the three house from the simplifies into finally, to X square terms the derivative of less X f of X minus born over two x to the three house.