Problems from Derivative of a power

Find the derivative of the following functions:

a)$$f (x) = x^{23}$$

b)$$f(x)=\sqrt{x^7}$$

c)$$f (x) =\sqrt[9]{x^4}$$

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Development:

a) The exponent is $$23$$. Therefore, $$f'(x) =23x^{23-1}=23x^{22}$$

b) $$f(x)=\sqrt{x^7}=x^{7/2}$$; In this case the exponent is $$7/2$$, and therefore $$f'(x)=\dfrac{7}{2}x^{5/2}$$.

c) $$f(x)=\sqrt[9]{x^4}=x^{4/9}$$. The exponent is $$4/9$$, and therefore $$f'(x)=\dfrac{4}{9}x^{-5/9}=\dfrac{4}{9\sqrt[9]{x^5}}$$.

Solution:

a) $$23x^{22}$$

b) $$\dfrac{7}{2}x^{5/2}$$

c) $$\dfrac{4}{9}x^{-5/9}=\dfrac{4}{9\sqrt[9]{x^5}}$$

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