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Calculus Examples
h(x)=(x+2)7-7x-1h(x)=(x+2)7−7x−1
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
By the Sum Rule, the derivative of (x+2)7-7x-1 with respect to x is ddx[(x+2)7]+ddx[-7x]+ddx[-1].
f′(x)=ddx((x+2)7)+ddx(-7x)+ddx(-1)
Step 1.1.2
Evaluate ddx[(x+2)7].
Step 1.1.2.1
Differentiate using the chain rule, which states that ddx[f(g(x))] is f′(g(x))g′(x) where f(x)=x7 and g(x)=x+2.
Step 1.1.2.1.1
To apply the Chain Rule, set u as x+2.
f′(x)=ddu(u7)ddx(x+2)+ddx(-7x)+ddx(-1)
Step 1.1.2.1.2
Differentiate using the Power Rule which states that ddu[un] is nun-1 where n=7.
f′(x)=7u6ddx(x+2)+ddx(-7x)+ddx(-1)
Step 1.1.2.1.3
Replace all occurrences of u with x+2.
f′(x)=7(x+2)6ddx(x+2)+ddx(-7x)+ddx(-1)
f′(x)=7(x+2)6ddx(x+2)+ddx(-7x)+ddx(-1)
Step 1.1.2.2
By the Sum Rule, the derivative of x+2 with respect to x is ddx[x]+ddx[2].
f′(x)=7(x+2)6(ddx(x)+ddx(2))+ddx(-7x)+ddx(-1)
Step 1.1.2.3
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
f′(x)=7(x+2)6(1+ddx(2))+ddx(-7x)+ddx(-1)
Step 1.1.2.4
Since 2 is constant with respect to x, the derivative of 2 with respect to x is 0.
f′(x)=7(x+2)6(1+0)+ddx(-7x)+ddx(-1)
Step 1.1.2.5
Add 1 and 0.
f′(x)=7(x+2)6⋅1+ddx(-7x)+ddx(-1)
Step 1.1.2.6
Multiply 7 by 1.
f′(x)=7(x+2)6+ddx(-7x)+ddx(-1)
f′(x)=7(x+2)6+ddx(-7x)+ddx(-1)
Step 1.1.3
Evaluate ddx[-7x].
Step 1.1.3.1
Since -7 is constant with respect to x, the derivative of -7x with respect to x is -7ddx[x].
f′(x)=7(x+2)6-7ddxx+ddx(-1)
Step 1.1.3.2
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
f′(x)=7(x+2)6-7⋅1+ddx(-1)
Step 1.1.3.3
Multiply -7 by 1.
f′(x)=7(x+2)6-7+ddx(-1)
f′(x)=7(x+2)6-7+ddx(-1)
Step 1.1.4
Differentiate using the Constant Rule.
Step 1.1.4.1
Since -1 is constant with respect to x, the derivative of -1 with respect to x is 0.
f′(x)=7(x+2)6-7+0
Step 1.1.4.2
Add 7(x+2)6-7 and 0.
f′(x)=7(x+2)6-7
f′(x)=7(x+2)6-7
f′(x)=7(x+2)6-7
Step 1.2
The first derivative of h(x) with respect to x is 7(x+2)6-7.
7(x+2)6-7
7(x+2)6-7
Step 2
Step 2.1
Set the first derivative equal to 0.
7(x+2)6-7=0
Step 2.2
Simplify 7(x+2)6-7.
Step 2.2.1
Simplify each term.
Step 2.2.1.1
Use the Binomial Theorem.
7(x6+6x5⋅2+15x4⋅22+20x3⋅23+15x2⋅24+6x⋅25+26)-7=0
Step 2.2.1.2
Simplify each term.
Step 2.2.1.2.1
Multiply 2 by 6.
7(x6+12x5+15x4⋅22+20x3⋅23+15x2⋅24+6x⋅25+26)-7=0
Step 2.2.1.2.2
Raise 2 to the power of 2.
7(x6+12x5+15x4⋅4+20x3⋅23+15x2⋅24+6x⋅25+26)-7=0
Step 2.2.1.2.3
Multiply 4 by 15.
7(x6+12x5+60x4+20x3⋅23+15x2⋅24+6x⋅25+26)-7=0
Step 2.2.1.2.4
Raise 2 to the power of 3.
7(x6+12x5+60x4+20x3⋅8+15x2⋅24+6x⋅25+26)-7=0
Step 2.2.1.2.5
Multiply 8 by 20.
7(x6+12x5+60x4+160x3+15x2⋅24+6x⋅25+26)-7=0
Step 2.2.1.2.6
Raise 2 to the power of 4.
7(x6+12x5+60x4+160x3+15x2⋅16+6x⋅25+26)-7=0
Step 2.2.1.2.7
Multiply 16 by 15.
7(x6+12x5+60x4+160x3+240x2+6x⋅25+26)-7=0
Step 2.2.1.2.8
Raise 2 to the power of 5.
7(x6+12x5+60x4+160x3+240x2+6x⋅32+26)-7=0
Step 2.2.1.2.9
Multiply 32 by 6.
7(x6+12x5+60x4+160x3+240x2+192x+26)-7=0
Step 2.2.1.2.10
Raise 2 to the power of 6.
7(x6+12x5+60x4+160x3+240x2+192x+64)-7=0
7(x6+12x5+60x4+160x3+240x2+192x+64)-7=0
Step 2.2.1.3
Apply the distributive property.
7x6+7(12x5)+7(60x4)+7(160x3)+7(240x2)+7(192x)+7⋅64-7=0
Step 2.2.1.4
Simplify.
Step 2.2.1.4.1
Multiply 12 by 7.
7x6+84x5+7(60x4)+7(160x3)+7(240x2)+7(192x)+7⋅64-7=0
Step 2.2.1.4.2
Multiply 60 by 7.
7x6+84x5+420x4+7(160x3)+7(240x2)+7(192x)+7⋅64-7=0
Step 2.2.1.4.3
Multiply 160 by 7.
7x6+84x5+420x4+1120x3+7(240x2)+7(192x)+7⋅64-7=0
Step 2.2.1.4.4
Multiply 240 by 7.
7x6+84x5+420x4+1120x3+1680x2+7(192x)+7⋅64-7=0
Step 2.2.1.4.5
Multiply 192 by 7.
7x6+84x5+420x4+1120x3+1680x2+1344x+7⋅64-7=0
Step 2.2.1.4.6
Multiply 7 by 64.
7x6+84x5+420x4+1120x3+1680x2+1344x+448-7=0
7x6+84x5+420x4+1120x3+1680x2+1344x+448-7=0
7x6+84x5+420x4+1120x3+1680x2+1344x+448-7=0
Step 2.2.2
Subtract 7 from 448.
7x6+84x5+420x4+1120x3+1680x2+1344x+441=0
7x6+84x5+420x4+1120x3+1680x2+1344x+441=0
Step 2.3
Graph each side of the equation. The solution is the x-value of the point of intersection.
x=-3,-1
x=-3,-1
Step 3
The values which make the derivative equal to 0 are -3,-1.
-3,-1
Step 4
Split (-∞,∞) into separate intervals around the x values that make the derivative 0 or undefined.
(-∞,-3)∪(-3,-1)∪(-1,∞)
Step 5
Step 5.1
Replace the variable x with -4 in the expression.
h′(-4)=7((-4)+2)6-7
Step 5.2
Simplify the result.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Add -4 and 2.
h′(-4)=7(-2)6-7
Step 5.2.1.2
Raise -2 to the power of 6.
h′(-4)=7⋅64-7
Step 5.2.1.3
Multiply 7 by 64.
h′(-4)=448-7
h′(-4)=448-7
Step 5.2.2
Subtract 7 from 448.
h′(-4)=441
Step 5.2.3
The final answer is 441.
441
441
Step 5.3
At x=-4 the derivative is 441. Since this is positive, the function is increasing on (-∞,-3).
Increasing on (-∞,-3) since h′(x)>0
Increasing on (-∞,-3) since h′(x)>0
Step 6
Step 6.1
Replace the variable x with -2 in the expression.
h′(-2)=7((-2)+2)6-7
Step 6.2
Simplify the result.
Step 6.2.1
Simplify each term.
Step 6.2.1.1
Add -2 and 2.
h′(-2)=7⋅06-7
Step 6.2.1.2
Raising 0 to any positive power yields 0.
h′(-2)=7⋅0-7
Step 6.2.1.3
Multiply 7 by 0.
h′(-2)=0-7
h′(-2)=0-7
Step 6.2.2
Subtract 7 from 0.
h′(-2)=-7
Step 6.2.3
The final answer is -7.
-7
-7
Step 6.3
At x=-2 the derivative is -7. Since this is negative, the function is decreasing on (-3,-1).
Decreasing on (-3,-1) since h′(x)<0
Decreasing on (-3,-1) since h′(x)<0
Step 7
Step 7.1
Replace the variable x with 0 in the expression.
h′(0)=7((0)+2)6-7
Step 7.2
Simplify the result.
Step 7.2.1
Simplify each term.
Step 7.2.1.1
Add 0 and 2.
h′(0)=7⋅26-7
Step 7.2.1.2
Raise 2 to the power of 6.
h′(0)=7⋅64-7
Step 7.2.1.3
Multiply 7 by 64.
h′(0)=448-7
h′(0)=448-7
Step 7.2.2
Subtract 7 from 448.
h′(0)=441
Step 7.2.3
The final answer is 441.
441
441
Step 7.3
At x=0 the derivative is 441. Since this is positive, the function is increasing on (-1,∞).
Increasing on (-1,∞) since h′(x)>0
Increasing on (-1,∞) since h′(x)>0
Step 8
List the intervals on which the function is increasing and decreasing.
Increasing on: (-∞,-3),(-1,∞)
Decreasing on: (-3,-1)
Step 9