Essential Knowledge 2.1C5

Essential Knowledge 2.1C5 Students will know that the chain rule is the basis for implicit differentiation.

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99% of equations in high school are written explicitly. This means that the variable of interest has already been isolated.

y = x²sin(x)

Here you can see y has already been isolated, so if we want to know y' we can start differentiating right away.

But some equations are written implicitly, meaning that a variable isn't isolated. A good example is the equation of a circle.

The equation for a circle centered at the origin and with a radius of 4 is

x² + y² = 16

Let's say we want to find the slope of a tangent line on the circle at a certain point. We could isolate y and then differentiate explicitly as we have been doing, but let's try doing some implicit differentiation just for practice.

Example

For the equation x² + y² = 16, determine

You see that the denominator is dx, so start by differentiating both sides by x. In other words, you just d/dx both sides.

The main thing to remember is to use the chain rule when necessary. Let's take this example slowly and look at what happens here.

As you can see, according to the sum rule we will have three separate derivatives.

The first term, , is simple, because we are differentiating with respect to x, and the only variable is x. No need for the chain rule here.

The second term, , is not as simple, because we are differentiating with respect to x, and the only variable is y. Here we would need the chain rule. Let's set it up for Leibniz's notation.

Remember, we need an equation of z in terms of y for our outside equation, and we need an equation of y in terms of x for our outside equation. Here's how we can do that.

z = y²

meaning now . Therefore we can answer our original question by finding dz/dx.

y = ? (we really just don't know what y could be equal to)

Now we need their derivatives, because our formula is

(since we don't know y, we certainly don't know dy/dx. That's ok, we'll just leave any answers as dy/dx. In fact, this is a good thing, because dy/dx is what we're supposed to be solving for!)

Now we can apply the formula

so now we have an answer for the 2nd term

The third term, , is simple. The derivative of a number is always 0.

Now let's substitute in all of the derivatives we've found thus far.

Now it's just a matter of isolating the variable we're interested in. The example instructed us to find dy/dx, so isolate dy/dx

And there's your answer.

We could have solved this by isolating y first. In some questions however, that is not an option, so it's still good to know how to differentiate implicitly using the chain rule.