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How do you find the derivative of #y = (1/3)^(x^2)#?

Answer 1

Emilia Aguilar

#dy/dx = -(2xln3)/(3^(x^2))#

#y= (1/3)^(x^2)#

Take the natural logarithm of both sides.

#lny = ln(1/3)^(x^2)#

Use laws of logarithms to simplify.

#lny = x^2ln(1/3)#

Differentiate using implicit differentiation and the product rule.

#1/y(dy/dx) = 2x(ln(1/3)) + x^2(0)#

#dy/dx= (2xln(1/3))/(1/y)#

#dy/dx = (1/3)^(x^2)2xln(1/3)#

#dy/dx = -2x(1/3)^(x^2)ln3 -> "since "ln(1/3) = ln(3^-1) = -ln3#

#dy/dx = -(2xln3)/(3^(x^2))#

Hopefully this helps!

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Answer 2

Isabella Ackerman

To find the derivative of ( y = (1/3)^{x^2} ), you can use the chain rule. First, take the natural logarithm of both sides to simplify the expression. Then, differentiate implicitly with respect to ( x ) and solve for ( \frac{dy}{dx} ). The derivative of ( y ) with respect to ( x ) is ( \frac{dy}{dx} = -2x (\ln(1/3)) (1/3)^{x^2} ).

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Answer from HIX Tutor

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.