# How do you differentiate #y=(e^(5x^4))/(e^(4x^2+3))#?

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To differentiate the function y=(e^(5x^4))/(e^(4x^2+3)), you can use the quotient rule of differentiation:

Let u = e^(5x^4) and v = e^(4x^2+3).

Then, y' = (u'v - uv') / v^2, where u' and v' are the derivatives of u and v with respect to x.

First, find the derivatives: u' = d(e^(5x^4))/dx = 20x^3 * e^(5x^4) v' = d(e^(4x^2+3))/dx = 8x * e^(4x^2+3)

Now substitute into the quotient rule formula: y' = (20x^3 * e^(5x^4) * e^(4x^2+3) - e^(5x^4) * 8x * e^(4x^2+3)) / (e^(4x^2+3))^2

Simplify: y' = (20x^3 * e^(9x^4+3) - 8x * e^(9x^4+3)) / e^(8x^2+6)

Factor out common terms: y' = (e^(9x^4+3) * (20x^3 - 8x)) / e^(8x^2+6)

Simplify further: y' = (20x^3 - 8x) * e^(9x^4+3 - 8x^2 - 6)

Finally: y' = (20x^3 - 8x) * e^(9x^4-8x^2-3)

That's the derivative of the function y=(e^(5x^4))/(e^(4x^2+3)).

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To differentiate the function ( y = \frac{e^{5x^4}}{e^{4x^2 + 3}} ), you can use the quotient rule.

The quotient rule states that if you have a function of the form ( \frac{u(x)}{v(x)} ), then its derivative is given by:

[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]

Applying this rule to the given function, where ( u(x) = e^{5x^4} ) and ( v(x) = e^{4x^2 + 3} ), and using the chain rule for differentiation, we have:

[ y' = \frac{(e^{5x^4})'(e^{4x^2 + 3}) - e^{5x^4}(e^{4x^2 + 3})'}{(e^{4x^2 + 3})^2} ]

[ = \frac{(5x^4)'e^{4x^2 + 3} - e^{5x^4}(8x)e^{4x^2 + 3}}{(e^{4x^2 + 3})^2} ]

[ = \frac{20x^3e^{4x^2 + 3} - 8xe^{9x^4 + 3}}{e^{2(4x^2 + 3)}} ]

[ = \frac{20x^3e^{4x^2 + 3} - 8xe^{4x^2 + 3}}{e^{8x^2 + 6}} ]

[ = \frac{4x^3(5 - 2x)e^{4x^2 + 3}}{e^{8x^2 + 6}} ]

So, the derivative of ( y ) with respect to ( x ) is ( y' = \frac{4x^3(5 - 2x)e^{4x^2 + 3}}{e^{8x^2 + 6}} ).

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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.

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