# How do you implicitly differentiate #-1=xy^2+x^2y-e^y-sec(xy) #?

Start with

Let's replace the secant with a cosine.

Now we take the derivative wrt x on BOTH SIDES!

The derivative of a constant is zero and the derivative is linear!

Now using product rule on just the first two terms we get!

Next lots and lots of Fun with the chain rule! Watch the last term!

(also doing the simple x derivatives )

Doing some of those y derivatives, xy derivatives and cos(xy) derivatives also doing the product rule and chain rule one more time on the last part of the last term.

Neaten a bit and finish all the derivatives

Now separate into term with

Bring everyting without

Divide though to find

That was very long!

Went with a VERY long explanation with simple example because implicit differentiation can be tricky and the chain rule is very very very important.

You need to use about three BIG Calculus rules to solve this and three specific function derivative.

You can keep going with this

The key to implicit differentiation is to use the chain rule to take the derivative wrt x of and function of both x and y, like a circle.

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To implicitly differentiate the equation ( -1 = xy^2 + x^2y - e^y - \sec(xy) ), you would differentiate each term with respect to ( x ) using the chain rule where necessary.

( \frac{d}{dx}(-1) = 0 )

( \frac{d}{dx}(xy^2) = y^2 + 2xy\frac{dy}{dx} )

( \frac{d}{dx}(x^2y) = x^2 + 2xy\frac{dx}{dx} )

( \frac{d}{dx}(-e^y) = -e^y\frac{dy}{dx} )

( \frac{d}{dx}(-\sec(xy)) = -\sec(xy)\tan(xy)(y+x\frac{dy}{dx}) )

So, the implicit differentiation of the given equation with respect to ( x ) yields:

( y^2 + 2xy\frac{dy}{dx} + x^2 + 2xy\frac{dx}{dx} - e^y\frac{dy}{dx} - \sec(xy)\tan(xy)(y+x\frac{dy}{dx}) = 0 )

Simplify the terms and solve for ( \frac{dy}{dx} ).

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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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