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How do you find the derivative of #z=x(y^2)-e^(xy)#?

Answer 1

Christopher Agresta

# (partial z) / (partial x) = y^2-ye^(xy) #
# (partial z) / (partial y) = 2xy-xe^(xy) #

We have:

#z=xy^2-e^(xy)#

Which is a function of two variables, so the derivatives are;

# (partial z) / (partial x) = y^2-ye^(xy) #

# (partial z) / (partial y) = 2xy-xe^(xy) #

Remember when partially differentiating: differentiate with respect to the variable in question, treating the other variables as constant.

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Answer 2

Isaiah Allen

To find the derivative of ( z = x(y^2) - e^{xy} ) with respect to ( x ), you use the product rule and chain rule.

First, apply the product rule to ( x(y^2) ) to get ( x ) times the derivative of ( y^2 ) plus ( y^2 ) times the derivative of ( x ). Then, apply the chain rule to ( e^{xy} ) to get the derivative of the outer function multiplied by the derivative of the inner function.

So, the derivative of ( z ) with respect to ( x ) is:

[ \frac{dz}{dx} = y^2 + x(2y \frac{dy}{dx}) - e^{xy}(y + x\frac{dy}{dx}) ]

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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.