In Exercises 21–48, find the derivative of y with respect to t...

Question

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

47. y=(arccot(x³))³

Accepted Answer

Welcome back everyone. Find the d y divided by dx given that y equals inverse of tangent of x 2, all erased to the power of 4. For this problem we once find the derivative of a composite function, meaning we once applied the chain rule. Let's remember that according to the chain rule we begin with the outer function. That's inverse of tangent. Of x 2 raised to the power of 4 and we want to differentiate it with respect to the first inner function. And that would be inverse of tension. Of X 2 According to the chain rule, we are multiplying by the derivative of the first inner function, so we are multiplying by the derivative of inverse of tangent of x squared. With respect to the second inner function, so with respect to x 2. And finally we are multiplying by the derivative of the 2nd inner function which is x squared. With respect to X. From here let's identify each derivative and let's notice that the first derivative requires us to use the power rule. We bring down the exponent, which is 4, and decrease the original exponent by 1 unit, so we get 4. Inverse of the ancient of x squared. Raise to the power of 3. For the second derivative we want to differentiate inverse of tangent. Let's remember that the derivative of inverse of tangent is 1 divided by. 1 plus the inner function squared, so that would be 1 divided by 1 + x 22. And finally, the derivative of x 2 is 2 x using the power rule. From here we can simplify the result. Let's begin with the numerator, that'll be 2 multiplied by 4, which is 8, followed by x, followed by inverse of tangent of x squared. Followed by cubed and in the denominator we have 1 + x raised to the power of 4 and that's our final answer for this problem. Thank you for watching.

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.47. y=(arccot(x³))³

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In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
47. y=(arccot(x³))³