Using the Power Rule

Find the derivative of:

1. \(y=x^{\sqrt{2}}\)
2. \(y=(x^{2}+1) ^{\pi}\)

Solution

1. We differentiate \(y\) using the Power Rule. \[ y^\prime =\frac{d}{dx}x^{\sqrt{2}}=\sqrt{2}x^{\sqrt{2}-1} \]
2. The Power Rule for Functions also holds when the exponent is a real number. Then \[ y^\prime =\frac{d}{dx}(x^{2}+1) ^{\pi }=\pi (x^{2}+1) ^{\pi -1}(2x) =2\pi x(x^{2}+1) ^{\pi-1} \]