Precalculus Examples

f(x)=x2+8x-4 , x=4
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)=(x+h)2+8(x+h)-4
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)=(x+h)(x+h)+8(x+h)-4
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)=x(x+h)+h(x+h)+8(x+h)-4
Step 2.1.2.1.2.2
Apply the distributive property.
f(x+h)=xx+xh+h(x+h)+8(x+h)-4
Step 2.1.2.1.2.3
Apply the distributive property.
f(x+h)=xx+xh+hx+hh+8(x+h)-4
f(x+h)=xx+xh+hx+hh+8(x+h)-4
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)=x2+xh+hx+hh+8(x+h)-4
Step 2.1.2.1.3.1.2
Multiply h by h.
f(x+h)=x2+xh+hx+h2+8(x+h)-4
f(x+h)=x2+xh+hx+h2+8(x+h)-4
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)=x2+hx+hx+h2+8(x+h)-4
Step 2.1.2.1.3.2.2
Add hx and hx.
f(x+h)=x2+2hx+h2+8(x+h)-4
f(x+h)=x2+2hx+h2+8(x+h)-4
f(x+h)=x2+2hx+h2+8(x+h)-4
Step 2.1.2.1.4
Apply the distributive property.
f(x+h)=x2+2hx+h2+8x+8h-4
f(x+h)=x2+2hx+h2+8x+8h-4
Step 2.1.2.2
The final answer is x2+2hx+h2+8x+8h-4.
x2+2hx+h2+8x+8h-4
x2+2hx+h2+8x+8h-4
x2+2hx+h2+8x+8h-4
Step 2.2
Reorder.
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Step 2.2.1
Move 8x.
x2+2hx+h2+8h+8x-4
Step 2.2.2
Move x2.
2hx+h2+x2+8h+8x-4
Step 2.2.3
Reorder 2hx and h2.
h2+2hx+x2+8h+8x-4
h2+2hx+x2+8h+8x-4
Step 2.3
Find the components of the definition.
f(x+h)=h2+2hx+x2+8h+8x-4
f(x)=x2+8x-4
f(x+h)=h2+2hx+x2+8h+8x-4
f(x)=x2+8x-4
Step 3
Plug in the components.
f(x+h)-f(x)h=h2+2hx+x2+8h+8x-4-(x2+8x-4)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.
h2+2hx+x2+8h+8x-4-x2-(8x)--4h
Step 4.1.2
Simplify.
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Step 4.1.2.1
Multiply 8 by -1.
h2+2hx+x2+8h+8x-4-x2-8x--4h
Step 4.1.2.2
Multiply -1 by -4.
h2+2hx+x2+8h+8x-4-x2-8x+4h
h2+2hx+x2+8h+8x-4-x2-8x+4h
Step 4.1.3
Subtract x2 from x2.
h2+2hx+8h+8x-4+0-8x+4h
Step 4.1.4
Add h2 and 0.
h2+2hx+8h+8x-4-8x+4h
Step 4.1.5
Subtract 8x from 8x.
h2+2hx+8h+0-4+4h
Step 4.1.6
Add h2 and 0.
h2+2hx+8h-4+4h
Step 4.1.7
Add -4 and 4.
h2+2hx+8h+0h
Step 4.1.8
Add h2+2hx+8h and 0.
h2+2hx+8hh
Step 4.1.9
Factor h out of h2+2hx+8h.
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Step 4.1.9.1
Factor h out of h2.
hh+2hx+8hh
Step 4.1.9.2
Factor h out of 2hx.
h(h)+h(2x)+8hh
Step 4.1.9.3
Factor h out of 8h.
h(h)+h(2x)+h8h
Step 4.1.9.4
Factor h out of h(h)+h(2x).
h(h+2x)+h8h
Step 4.1.9.5
Factor h out of h(h+2x)+h8.
h(h+2x+8)h
h(h+2x+8)h
h(h+2x+8)h
Step 4.2
Reduce the expression by cancelling the common factors.
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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.
h(h+2x+8)h
Step 4.2.1.2
Divide h+2x+8 by 1.
h+2x+8
h+2x+8
Step 4.2.2
Reorder h and 2x.
2x+h+8
2x+h+8
2x+h+8
Step 5
Replace the variable x with 4 in the expression.
2(4)+h+8
Step 6
Multiply 2 by 4.
8+h+8
Step 7
Add 8 and 8.
h+16
Step 8
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