Precalculus Examples

Find the Difference Quotient
f(x)=12x2-3
Step 1
Consider the difference quotient formula.
f(x+h)-f(x)h
Step 2
Find the components of the definition.
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Step 2.1
Evaluate the function at x=x+h.
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Step 2.1.1
Replace the variable x with x+h in the expression.
f(x+h)=12(x+h)2-3
Step 2.1.2
Simplify the result.
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Step 2.1.2.1
Simplify each term.
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Step 2.1.2.1.1
Rewrite (x+h)2 as (x+h)(x+h).
f(x+h)=12((x+h)(x+h))-3
Step 2.1.2.1.2
Expand (x+h)(x+h) using the FOIL Method.
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Step 2.1.2.1.2.1
Apply the distributive property.
f(x+h)=12(x(x+h)+h(x+h))-3
Step 2.1.2.1.2.2
Apply the distributive property.
f(x+h)=12(xx+xh+h(x+h))-3
Step 2.1.2.1.2.3
Apply the distributive property.
f(x+h)=12(xx+xh+hx+hh)-3
f(x+h)=12(xx+xh+hx+hh)-3
Step 2.1.2.1.3
Simplify and combine like terms.
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Step 2.1.2.1.3.1
Simplify each term.
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Step 2.1.2.1.3.1.1
Multiply x by x.
f(x+h)=12(x2+xh+hx+hh)-3
Step 2.1.2.1.3.1.2
Multiply h by h.
f(x+h)=12(x2+xh+hx+h2)-3
f(x+h)=12(x2+xh+hx+h2)-3
Step 2.1.2.1.3.2
Add xh and hx.
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Step 2.1.2.1.3.2.1
Reorder x and h.
f(x+h)=12(x2+hx+hx+h2)-3
Step 2.1.2.1.3.2.2
Add hx and hx.
f(x+h)=12(x2+2hx+h2)-3
f(x+h)=12(x2+2hx+h2)-3
f(x+h)=12(x2+2hx+h2)-3
Step 2.1.2.1.4
Apply the distributive property.
f(x+h)=12x2+12(2hx)+12h2-3
Step 2.1.2.1.5
Multiply 2 by 12.
f(x+h)=12x2+24hx+12h2-3
f(x+h)=12x2+24hx+12h2-3
Step 2.1.2.2
The final answer is 12x2+24hx+12h2-3.
12x2+24hx+12h2-3
12x2+24hx+12h2-3
12x2+24hx+12h2-3
Step 2.2
Reorder.
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Step 2.2.1
Move 12x2.
24hx+12h2+12x2-3
Step 2.2.2
Reorder 24hx and 12h2.
12h2+24hx+12x2-3
12h2+24hx+12x2-3
Step 2.3
Find the components of the definition.
f(x+h)=12h2+24hx+12x2-3
f(x)=12x2-3
f(x+h)=12h2+24hx+12x2-3
f(x)=12x2-3
Step 3
Plug in the components.
f(x+h)-f(x)h=12h2+24hx+12x2-3-(12x2-3)h
Step 4
Simplify.
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Step 4.1
Simplify the numerator.
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Step 4.1.1
Apply the distributive property.
12h2+24hx+12x2-3-(12x2)--3h
Step 4.1.2
Multiply 12 by -1.
12h2+24hx+12x2-3-12x2--3h
Step 4.1.3
Multiply -1 by -3.
12h2+24hx+12x2-3-12x2+3h
Step 4.1.4
Subtract 12x2 from 12x2.
12h2+24hx+0-3+3h
Step 4.1.5
Add 12h2 and 0.
12h2+24hx-3+3h
Step 4.1.6
Add -3 and 3.
12h2+24hx+0h
Step 4.1.7
Add 12h2+24hx and 0.
12h2+24hxh
Step 4.1.8
Factor 12h out of 12h2+24hx.
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Step 4.1.8.1
Factor 12h out of 12h2.
12hh+24hxh
Step 4.1.8.2
Factor 12h out of 24hx.
12hh+12h(2x)h
Step 4.1.8.3
Factor 12h out of 12hh+12h(2x).
12h(h+2x)h
12h(h+2x)h
12h(h+2x)h
Step 4.2
Simplify terms.
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Step 4.2.1
Cancel the common factor of h.
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Step 4.2.1.1
Cancel the common factor.
12h(h+2x)h
Step 4.2.1.2
Divide 12(h+2x) by 1.
12(h+2x)
12(h+2x)
Step 4.2.2
Apply the distributive property.
12h+12(2x)
Step 4.2.3
Simplify the expression.
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Step 4.2.3.1
Multiply 2 by 12.
12h+24x
Step 4.2.3.2
Reorder 12h and 24x.
24x+12h
24x+12h
24x+12h
24x+12h
Step 5
Enter YOUR Problem
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 [x2  12  π  xdx ] 
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