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Find the derivatives of the given functions.$$r=0.5\left(\sin ^{-1} 3 t\right)^{4}$$

$\frac{d u}{d t}=\frac{8 \sin ^{-1}(4 t+3)}{\sqrt{1-(4 t+3)^{2}}}$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 3

Derivatives of the Inverse Trigonometric Functions

Derivatives

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we want to find do I. D. T. For the expression why is equal to 0.5 times the arc sine of three T raised to the fourth power. Just all this problem, we're going to rely on derivative shortcuts we picked up throughout single variable calculus in addition to the inverse trig derivatives that we've recently learned. So I've listed here the major inverse trig derivatives were going to primarily rely on the fact that D. D. X. Marks an active route 1 1/1 minus X square. We're also gonna need to use the chain rule since our arc sine is raised to the power of four and has argument three T. We note that why is already written in the most easily differential forms we perceive solved by the chain rule we have DY DT equals 0.5 times four times are three T. Cute times the rhythm of what's inside which is three over route one minus 90 squared, Multiplying out our constants and simplifying the expression gives DY DT equals six arc sine three T. Cute over route one minus 90 squared

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

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