Evaluate and simplify y'.
sin x cos(y−1) = 1/2

Question

Evaluate and simplify y'.

sin x cos(y−1) = 1/2

Accepted Answer

Welcome back, everyone. Evaluate and simplify the derivative C for the equation cosine of T multiplied by sine of Z + 2 equals V4s were given for answer choices. A says Z equals negative tangent of T multiplied by tangent of Z + 2. B negative cotangent of T multiplied by cotangent of C + 2. C cotangent of T multiplied by cotangent of C + 2 and D tangent oft multiplied by tangent of C + 2. So for this problem, let's understand that Z is simply a function of T. So Z equals Z of T. And we want to slur C, so we're going to apply implicit differentiation. This means that we're taking the derivative of the left hand side where we have a product. cosine of T is multiplied by sine of Z + 2. We're taking the derivative of the left hand side and setting it equal to the derivative of the right hand side. On the left hand side we have a product, so we have to take the derivative of the first function, cosine oft derivative multiplied by sign of Z + 2 based on the product rule, and then we're adding the first function cosine of T multiplied by the derivative of the second function. So the derivative of sine of Z + 2. Here we have our product rule and then we're setting it equal to the derivative of 3/4, which is 0 because it is a constant. Now, the derivative of cosine of T is negative sine of T based on the table of basic derivatives. We're multiplying that by sine of Z + 2. And then we have to evaluate the derivative of sign. Of Z + 2, right? So we're multiplying cosine of T by the derivative. The derivative of sine is cosine. So we are essentially multiplying by cosine of C + 2. But now the inner function is a composite function. This means that applying the chain rule, we also have to multiply by the derivative of Z + 2. So let's simplify the expression. We're going to set it equal to 0 because on the right we have 0, and simplifying, we're going to get negative sign of T. Sign of Z + 2. Now, what is the derivative of Z + 2? Well, C is a function of T, so we're going to have C + 0 because the derivative of 2 is 0. So we can say that we're adding. Z Cosine of T multiplied by cosine of Z + 2. This is equal to 0, and now we can move the negative term to the right, which gives us Z. Cosine of T. Cosine of C + 2. Equals Sign of tea Sign of Z + 2. Now solving for C, we're going to divide the right hand side, which is sign of T. Sign of Z + 2 by the left hand side, specifically the factor adjacent to Z, which is cosine of T, cosine of Z + 2. And now let's recall that sine divided by cosine gives us tangent. So we have tangent of T. For the first ratio. Multiplied by ancient. Of the other angle which is Z + 2. And this is our final answer. Essentially our Z is equal to tangent of T multiplied by tangent of Z + 2, which corresponds to the answer choice D. Thank you for watching.

Evaluate and simplify y'.
sin x cos(y−1) = 1/2

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Identities

Watch next

Evaluate and simplify y'.sin x cos(y−1) = 1/2

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Identities

Watch next

Evaluate and simplify y'.
sin x cos(y−1) = 1/2