Calc 1 Homework ( pls only math experts)

Written Homework 2, Math122B section 5.

SHOW ALL WORK/ REASONINGS AND USE PROPER NOTATION FOR

FULL CREDIT.

During lecture, we proved the power rule for xn where n is a positive integer. We generalized

without proof that the power rule will persist for any real number n. In this homework, we will

use derivative rules and implicit differentiation to justify the power rule for non-positive-integer

power.

1 If the power of the power function is a rational number, we can write the power as m

n . Let

y = x m n . Then yn = xm. Use implicit differentiation to compute

dy

dx and confirm that the

power rule holds for rational power.

2 Suppose the power n is an irrational number. Let y = xn where n is irrational. Apply the

natural log to the equation to ”pull down” the exponent n. Then use implicit differentiation

to compute dy

dx and again, confirm that the power rule holds for irrational power.

3 The remaining case to check is negative power. Let y = x−n where n > 0. Use chain rule

to compute dy

dx and that the power rule holds for negative power as well.

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