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Chapter 6: Problem 91

Find the sixth derivative of \(\mathrm{y}=\mathrm{x}^{6}\).

Short Answer

Expert verified
The sixth derivative of \(y = x^6\) is \(y^{(6)} = 720\).

Step by step solution

01

Apply the First Derivative

To find the first derivative of y = x^6, use the power rule of differentiation: \(\frac{d}{dx} (x^n) = nx^{(n - 1)}\). Applying the power rule to x^6: \(y' = \frac{d}{dx} (x^6) = 6x^{(6 - 1)} = 6x^5\)
02

Apply the Second Derivative

Apply the power rule again to find the second derivative of y: \(y'' = \frac{d}{dx} (6x^5) = 30x^{(5 - 1)} = 30x^4\)
03

Apply the Third Derivative

Apply the power rule again to find the third derivative of y: \(y''' = \frac{d}{dx} (30x^4) = 120x^{(4 - 1)} = 120x^3\)
04

Apply the Fourth Derivative

Apply the power rule again to find the fourth derivative of y: \(y^{(4)} = \frac{d}{dx} (120x^3) = 360x^{(3 - 1)} = 360x^2\)
05

Apply the Fifth Derivative

Apply the power rule again to find the fifth derivative of y: \(y^{(5)} = \frac{d}{dx} (360x^2) = 720x^{(2 - 1)} = 720x\)
06

Apply the Sixth Derivative

Finally, apply the power rule one last time to find the sixth derivative of y: \(y^{(6)} = \frac{d}{dx} (720x) = 720\) The sixth derivative of y = x^6 is 720.

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