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Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the h0definition of the derivative and (b) the zxdefinition of the derivative.

fx=x3

Short Answer

Expert verified

f'x=3x2

Step by step solution

01

Given Information

The given expression isfx=x3

02

Simplification

Using (a)

f'x=limh0fx+h-fxh

f'x=limh0x+h3-x3h

role="math" localid="1650970063541" f'x=limh0x3+h3+3x2h+3xh2-x3h

role="math" localid="1650970073601" f'x=limh0hh2+3x2+3xhh

role="math" localid="1650970082470" f'x=limh0h2+3x2+2xh

Applying limit

f(x)=2x2

Using (b)

role="math" localid="1650970131635" f'x=limzxfz-fxz-x

role="math" localid="1650969972120" f'x=limzxz3-x3z-x

role="math" localid="1650969988908" f'x=limzxz-xz2+x2+2zxz-x

f'x=limzxz2+x2+2zx

Applying limit

f'x=3x2

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