Evaluate the derivative of the following functions.
f(y)...

Question

Evaluate the derivative of the following functions.

f(y) = tan^-1(2y² - 4)

Accepted Answer

Welcome back everyone. Find the derivative of g of T equals inverse of tangent of 4 t to the power of 6 minus 8. We are given 4 answer choices a 16 T to the power of worth divided by square root of 1 minus or T to the power of 6 minus 82 B. 16 to the power of 4th divided by square root of 1 minus 4 t to the power of 6 minus 82 C-24t power of 5th divided by 1 + 46 minus 82 and D 24 of 5th divided by 1 + 4 T 6 minus 82. So let's evaluate the derivative of G of T. And as we can see, we're evaluating the derivative of the inverse of 10 function, but we're also given the inner function, which means that we want to apply the chain rule. So first of all, we're going to show that we're evaluating the derivative of inverts of tangent of 40 to the power of 6 minus 8. And now we're going to expand this based on the chain rule. First of all, we're evaluating the derivative D. Divided by D. Of 40 to the power of 6 minus 8 of inverse of tangent. Of 4 t to the power of 6 minus 8, so we're essentially keeping our angle fixed in evaluating the derivative of inverts of tangent with respect to that angle and then based on the chain rule we have to multiply that by d divided by dt, so we're evaluating the derivative of the inner function with respect to T so. Words the power of 6 minus 8 is the final derivative that we want to get. Now for the first derivative we have to recall that the derivative of inverse of tangent of X based on the table of basic derivatives is 1 divided by 1 + x squared. So in this case we get 1 divided by 1 plus the given angle squared, which is for t to the power of 6 minus 82. And then we multiply by the derivative of 4 t to the power of 6. We can factor out the constant, that's 4, and then we apply the power rules, so we bring down the 6 and multiply by t to the power of 6 minus 1, which is 5. So we decrease the original exponent by 1 unit, and this gives us 24 t to the power of fifth, and of course the derivative of negative 8 is 0. So now we can just combine those on a single fraction, which gives us 24t the power of fifth divided by 1 + 4 T the power of 6 minus 8 squad and essentially this is our final answer. This final answer corresponds to the answer choice D. Thank you for watching.

Evaluate the derivative of the following functions.
f(y) = tan^-1(2y² - 4)

Key Concepts

Derivative

Chain Rule

Inverse Trigonometric Functions

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