Written Homework 2 Math122b Section 5show All Work Reasonings ✓ Solved

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 = xm/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 the chain rule to compute dy/dx and demonstrate that the power rule holds for negative power as well.

Paper For Above Instructions

The power rule is a fundamental concept in calculus that enables the differentiation of polynomial expressions efficiently. In this paper, we will explore the validity of the power rule for non-positive-integer powers using implicit differentiation and the properties of logarithms.

Rational Powers

Let us start with rational powers. Suppose we have the function y = xm/n, where m and n are integers with n > 0. First, rearranging gives us yn = xm. Now, we can differentiate both sides with respect to x using implicit differentiation:

n  yn-1  (dy/dx) = m * xm-1

Now, solving for dy/dx gives us:

dy/dx = (m/n) * x(m/n)-1

This result confirms that the power rule holds for rational powers because:

dy/dx = (1/n)  xm/n - 1 = (m/n)  x(m/n)-1

Irrational Powers

Next, consider when n is an irrational number, thus, let y = xn. We can apply the natural logarithm to express the exponent:

ln(y) = n * ln(x)

By differentiating both sides, we obtain:

1/y * (dy/dx) = n/x

Thus, solving for dy/dx yields:

dy/dx = y  (n/x) = xn  (n/x) = n * x(n-1)

This verification affirms that the power rule holds for irrational powers since dy/dx = n * x(n-1) works seamlessly.

Negative Powers

Lastly, we check for negative powers. Suppose y = x-n with n > 0. Using the chain rule assists in differentiating:

dy/dx = -n * x(-n-1)

This form is consistent with the established power rule, which maintains itself even when n is negative. Essentially:

dy/dx = -n  x-n-1 = -n  x(-n-1)

This indicates that the derivative rules we applied systematically uphold the integrity of the power rule across both irrational and negative numbers, thus extending its validity beyond simply positive integer powers.

Conclusion

Through the application of implicit differentiation and logarithmic manipulation, we have successfully confirmed that the power rule holds for rational, irrational, and negative powers. Such consistency in derivatives accentuates the critical role the power rule plays within calculus and reinforces our comprehension of differentiation standards critically.

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