9–61. Evaluate and simplify y'.

y = x²+2...

Question

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot x)

Accepted Answer

Welcome back, everyone. Calculate the derivative of the function Y equals 4x the 4th minus 4x inverse of cosine of of x. So for this function, let's write down the expression of the derivative. Y is going to be equal to 4. We can factor out the constant. Multiplied by the derivative of X the power of 4th minus 4, once can we factor out the constant multiplied by the derivative of X inverse of cosine of sine of x. And now what we want to do is evaluate two derivatives. The derivative of X, the power of 4th, can be found using the power rule. We bring down the 4. The original exponent decreases by 1 unit, so we get 4 X cubed. And for the second derivative, we have a product of two functions X and the inverse of cosine function, right? So what we want to do is simply recall that the inverse of cosine of x derivative is equal to -1 divided by square root of 1 minus X2, and we're going to use this definition in our example. So If we write the overall derivative that we want to calculate, we get X multiplied by the inverse of cosine of sine of x derivative. Now, based on the product rule, we have to recall that. We first will take the derivative of the first functions. We take the derivative of X, multiply that by the second function, which is inverse of cosine. Of sine of x and then we want to add the first function which is X multiplied by the derivative of the second function. So the derivative of the inverse of cosine of sine of x. Now the derivative of X is d x divided by d X, which is 1, so we end up with inverse of cosine of sine of x, and then we are adding X multiplied by the derivative of inverse of cosine of sine of x. So now based on the definition, we get -1 divided by square root of. 1 minus x squad, but in this case our angle is sin x, so we're subtracting sine squared of X, but because the inner function is not X, we have to apply the chain rule and multiply by the derivative of the inner function. So the derivative of sine of X. And now let's evaluate the remaining derivatives. We know that that the derivative of sin of X is cosine of X based on the table of basic derivatives, and we're going to get Inverts of cosine of sine of x. Minus And we're multiplying x by cosine of X, we get minus X, cosine of x divided by, and we have to recall based on the Pythagorean identity that 1 minus sine square x gives us cosine square x. So we have square root. Of cosine squared X. Which is the absolute value of cosine of x. Well then, we have successfully evaluated the final derivative, and now we can substitute the results into y. So we get 4. We can factor that because it is the repeating factor multiplied by the derivative of X, the power of 4th, which is 4 x cubed, and then we are subtracting inverse of cosine of sine of x. Right? And because we are subtracting, well, essentially subtracting the negative term will give us the positive term. So we are adding X cosine of x divided by the absolute value of cosine of X, and from here we can distribute the terms. So we're going to get 16 X cubed. -4. Inverse of cosine. Of sign of X. And then plus. Or X cosine X. Divided by the absolute value of cosine of x. And that will be our final answer, so let's label it and thank you for watching.

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot x)

Key Concepts

Differentiation

Chain Rule

Product Rule

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