Find the derivative of the following functions:
a)$$f (x) = x^{23}$$
b)$$f(x)=\sqrt{x^7}$$
c)$$f (x) =\sqrt[9]{x^4}$$
See development and solution
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) $$\dfrac{4}{9}x^{-5/9}=\dfrac{4}{9\sqrt[9]{x^5}}$$