(a) Find y^' by implicit differentiation. (b) Solve the...

(a) Find $y^{\prime}$ by implicit differentiation. (b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x .$ (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a). $\cos x+\sqrt{y}=5$

(a) Find $y^{\prime}$ by implicit differentiation.
(b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in
terms of $x .$
(c) Check that your solutions to parts (a) and (b) are consistent by
substituting the expression for $y$ into your solution for part (a).
$\cos x+\sqrt{y}=5$

Key Concepts

Explicit Differentiation

Explicit differentiation involves first solving the given equation for the dependent variable in terms of the independent variable. Once the equation is restructured into an explicit form, standard differentiation techniques can be applied directly. This approach often simplifies the differentiation process and can be used to verify results obtained through implicit differentiation.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent variable cannot be easily separated from the independent variable. This method involves taking the derivative of both sides of an equation with respect to the independent variable, treating the dependent variable as an implicit function. It also requires the use of the chain rule when differentiating terms that involve the dependent variable.

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. When a function is composed of an inner function and an outer function, the derivative is found by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function. This rule is essential in implicit differentiation, especially when differentiating terms like the square root or other composite functions.

Consistency Check

A consistency check is performed by confirming that different methods of obtaining the derivative produce equivalent results. In problems involving both implicit and explicit differentiation, substituting the expression for the dependent variable from the explicit solution into the result from implicit differentiation serves as a verification step to ensure that both methods yield the same derivative relationship, thereby validating the correctness of the solutions.

Transcript

00:01 We have the function cosine x plus square root of y equals 5.

00:07 And we want to find y prime in two different ways and check that they agree with each other.

00:16 So the first way is by implicit differentiation, which means that i'm going to take the derivative of both sides of this equation with respect to x, keeping in mind that y is a function of x.

00:28 So first we start with on the left side cosine x the derivative of cosine x is minus sine x plus.

00:38 Now i have the square root of y, which is really y to the one -half.

00:45 So by the chain rule, i first take the derivative of the outside function, which is raising an input to the power of one -half.

00:52 So by the power rule, i get that the derivative is one -half y to the minus one -half.

00:57 Then i have to multiply by the derivative of the inside function, which is y, and the derivative of that is y prime.

01:05 And this equals the derivative of 5 with respect to x, which is 0.

01:09 So now i am going to isolate the term that has y prime as a factor.

01:24 So 1 half y to the minus 1 half times y prime equals sine x.

01:29 And this is really another way to write this is y prime divided by two square root of y equals sine x and now it might be a little easier to see how to solve for y prime sine x or two sine x times square root of y now the second way that we can find y prime is first by solving for y from the original equation.

02:07 So we can move the cosine x over to the right.

02:11 So we get 5 minus cosine x.

02:13 And we can square everything and get, let's see, 25 minus 10 cosine x plus cosine squared x...

Please give Ace some feedback