Derivative of a Sum (ax^n)

Just like we saw in derivatives of a sum of the type x^n, 

So we can see that the derivative of the sum is the same as the sums of the derivatives. 

And this applies to any function, whether it be the $x^n$xn we saw before, or $ax^n$axn that we are exploring now.  

Examples

Find the derivative of the following, 

a) $f(x)=4x^2+3x+2$f(x)=4x2+3x+2, then $f'(x)=8x+3$f′(x)=8x+3   (remember that the derivative of a constant term is $0$0)

b) $f(x)=3x^3-3x^2$f(x)=3x3−3x2,  then $f'(x)=9x^2-6x$f′(x)=9x2−6x

c) $f(x)=6x^{-3}-2x+\sqrt{x}$f(x)=6x−3−2x+√x.  Firstly we need to turn the $\sqrt{x}$√x into a power.  $\sqrt{x}=x^{\frac{1}{2}}$√x=x12​ So $f(x)=6x^{-3}-2x+x^{\frac{1}{2}}$f(x)=6x−3−2x+x12​ and so then the derivative $f'(x)=-18x^{-4}-2+\frac{1}{2}x^{-\frac{1}{2}}$f′(x)=−18x−4−2+12​x−12​

Worked Examples:

question 1

question 2

question 3