Differentiating Implicitly

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Question

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

xy = cot(xy)

Accepted Answer

Welcome back, everyone. Use simplicity differentiation to find DV divided by Du for the equation UV equals tangent of UV. We're given for answer choices A says negative V divided by U B V divided by U, C U divided by V, and D negative U divided by V. So let's differentiate the left hand side. We get UV and the right hand side. We get the derivative of tangent of UV. On the left hand side we get UV derivative, and for the right hand side we can apply the chain rule. The derivative of tangent is 2 squared, the angle is UV. And according to the chain rule, we have to multiply that. By the derivative of the inner function, which is the derivative of UV. And now what we notice is that if UV is not equal to 0, we can just cancel it out, right, and end up with 1 equals 2 squared of UV. In other words, we're dividing both sides by UV, which is not equal to 0. So this equation holds if you multiplied by the derivative is not equal to 0. That's equation number one, and actually we will not need to use it because it doesn't contain EUV prime. The second case Is that UV prime. is equal to 0. Now if UV is equal to 0, which is reasonable because if we substitute 0 on the left and on the right, we're going to get 0 equals 0. So it is the solution of the original equation, right? And we are going to solve this equation. Let's apply the product rule. According to the product rule, we're going to get U V plus UV equals 0. And we know that V is a function of U. That's because U is in the denominator, so U is the independent variable, and the derivative of U is simply equal to 1 because it's the independent variable. We can then simplify the equation which gives us a V + U V equals 0, and we are solving for V, which means that UV equals negative V. We are subtracting V from both sides and our DV divided by D U. Which is is going to be equal to negative V. The right hand side of the result on the equation divided by the coefficient U, which is variable, right, in this case. So we have shown that DV divided by D U is equal to negative V divided by U. Which corresponds to the answer choice a. Thank you for watching.

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

xy = cot(xy)

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Derivatives

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