How do you differentiate #y=tanx/(2x^3)#?
If you are studying maths, then you should learn the Quotient Rule for Differentiation, and practice how to use it:
I was taught to remember the rule in word; " vdu minus udv all over v squared ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.
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To differentiate ( y = \frac{\tan(x)}{2x^3} ), you can use the quotient rule. The quotient rule states that if ( y = \frac{u}{v} ), where ( u ) and ( v ) are functions of ( x ), then the derivative of ( y ) with respect to ( x ) is given by:
[ \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ]
So, applying the quotient rule to the given function:
[ u = \tan(x) ] [ \frac{du}{dx} = \sec^2(x) ]
[ v = 2x^3 ] [ \frac{dv}{dx} = 6x^2 ]
Now, plug these into the quotient rule formula:
[ \frac{dy}{dx} = \frac{(2x^3)(\sec^2(x)) - (\tan(x))(6x^2)}{(2x^3)^2} ]
[ \frac{dy}{dx} = \frac{2x^3\sec^2(x) - 6x^2\tan(x)}{4x^6} ]
[ \frac{dy}{dx} = \frac{2\sec^2(x) - 6x\tan(x)}{4x^4} ]
This is the derivative of ( y ) with respect to ( x ).
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To differentiate ( y = \frac{\tan(x)}{2x^3} ) with respect to ( x ), you can use the quotient rule for differentiation:
If ( y = \frac{u}{v} ), then ( y' = \frac{u'v - uv'}{v^2} ),
where ( u' ) is the derivative of ( u ) with respect to ( x ), and ( v' ) is the derivative of ( v ) with respect to ( x ).
Applying the quotient rule to ( y = \frac{\tan(x)}{2x^3} ), we get:
( u = \tan(x) ) and ( v = 2x^3 ),
( u' = \sec^2(x) ) and ( v' = 6x^2 ).
Now, plug these values into the quotient rule formula:
( y' = \frac{\sec^2(x) \cdot 2x^3 - \tan(x) \cdot 6x^2}{(2x^3)^2} ).
Simplify this expression to get the derivative of ( y ) with respect to ( x ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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