How do you find #(dy)/(dx)# given #root3x+root3(y^4)=2#?
Start by writing the roots as rational exponents:
Differentiate each term using the power rule:
If you want it in radical form instead of rational exponent form:
Hopefully this helps!
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To find (\frac{{dy}}{{dx}}), we differentiate the equation (\sqrt{3x} + \sqrt{3y^4} = 2) implicitly with respect to (x) using the chain rule and product rule. After differentiation, we solve for (\frac{{dy}}{{dx}}).
Differentiating (\sqrt{3x} + \sqrt{3y^4} = 2) with respect to (x) yields:
[ \frac{1}{2\sqrt{3x}} \cdot 3 + \frac{1}{2\sqrt{3y^4}} \cdot 12y^3 \cdot \frac{{dy}}{{dx}} = 0 ]
Simplify the equation:
[ \frac{1}{{2\sqrt{3x}}} + \frac{{6y^3}}{{2\sqrt{3}y^2}}\frac{{dy}}{{dx}} = 0 ]
Now, solve for (\frac{{dy}}{{dx}}):
[ \frac{{dy}}{{dx}} = -\frac{1}{{6y^3}} \cdot \frac{{\sqrt{3x}}}{{\sqrt{3}}} ]
[ \frac{{dy}}{{dx}} = -\frac{1}{{6\sqrt{3}y^3}} \cdot \sqrt{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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