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Use the zxdefinition of the derivative to show that role="math" localid="1648386605099" ddx(x8)=8x7.

Short Answer

Expert verified

limzxz8-x8z-x=limzx(z-x)(z7+xz6+x2z5+x3z4+x4z3+x5z2+x6z+x7)z-x=limzx(z7+xz6+x2z5+x3z4+x4z3+x5z2+x6z+x7)=x7+x7+x7+x7+x7+x7+x7+x7=8x7.

Step by step solution

01

Step 1. Given Information 

We have given the following function :-

x8.

We have to use the zxdefinition of derivative, to prove that :-

ddx(x8)=8x7

02

Step 2. Derivative of given function :- 

Consider the given function as :-

f(x)=x8.

We know that zxdefinition of derivative is stated that :-

ddxf(x)=limzxf(z)-f(x)z-x

Put the values :-

role="math" localid="1648386886563" ddx(x8)=limzxz8-x8z-x

Simplify it :-

role="math" localid="1648386910268" ddx(x8)limzxz8-x8z-xddx(x8)=limzx(z-x)(z7+xz6+x2z5+x3z4+x4z3+x5z2+x6z+x7)z-xddx(x8)=limzx(z7+xz6+x2z5+x3z4+x4z3+x5z2+x6z+x7)ddx(x8)=x7+x7+x7+x7+x7+x7+x7+x7ddx(x8)=8x7

Hence proved.

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