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

Question

Evaluate the derivative of the following functions.

f(t) = ln (sin^-1 t²)

Accepted Answer

Welcome back, everyone. Find the derivative of H X equals LN of inverse of cosine of X to the power of 6. So for this problem we want to evaluate the derivative of H X. Let's call it H X. And essentially what we want to do is simply evaluate the derivative of LN of inverse of cosine of X, the power of 6. And what we have to recall is that this is a composite function. It's not LN of X, it's LN of a more complex function. So, let's recall that the derivative of LN of X is equal to 1 divided by X. We're going to apply the chain rule, right? First of all, we want to make sure that our angle is the inverse of cosine. So, based on the given rule, The first part of differentiation involves a 1 divided by the given angle, which is inverse of cosine of x the power of 6. But because this is a composite function, we then need to multiply this by the derivative of the inverts of cosine, right? So essentially we're going to write D divided by DX indicating that we are evaluating the derivative. But now because inverse of cosine has an angle that is also another function x the power of 6th, we're going to differentiate with respect to x to the power of 6th first. This is what the chain rule says, and we're going to differentiate the inverse of cosine of x to the power of 6. And we have to recall that the derivative of inverse of cosine of X is equal to -1 divided by square root of 1 minus X2. And then we're still not done because the final inner function is X, the power of 6. So eventually applying the chain rule again, we want to differentiate D. Divided by the X of X is the power of 6, right? This is our final function and here we're going to apply the power rule. So let's go ahead and simplify. So first of all, we get 1 divided by the inverse of cosine of X to the power of 6. Now we want to find the derivative of the inverse of cosine of X to the power of 6. We already know that this is -1. Divided by square root of 1 minus X2, right, but in this case our angle is x to the power of 6, so when we square it we get X to the power of 12th. And finally, we want to differentiate X to the power of 6 using the power rules. So x the power of N is equal to N multiplied by x the power of n minus 1. So we bring down the exponent and the original exponent decreases by 1 unit. Overall, we're going to get negative, right? We can factor out the negative sign. We can write 6 X to the power of fifth in the numerator. And then our denominator involves inverse of cosine. Of x the power of 6 multiplied by square root of 1 minus x to the power of 12, which is our final answer. Thank you for watching.

Evaluate the derivative of the following functions.
f(t) = ln (sin^-1 t²)

Key Concepts

Derivative

Chain Rule

Inverse Functions

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