12.4 - calculate the derivatives of a functions (polynomial, exponential, logarithmic)
d/du [a^u] is?
a^ulna
multiplication by a constant derivative, cf?
cf'
derivative of sin(x)?
cosx
y=log₃3x², apply chain rule, set u as 3x²...?
d/du[log₃(u)×d/dx[3x²]
Power Rule
d/dx x^n = nx^n-1
y=-3x+9, differentiate using the sum rule?
d/dx[-3x]+d/dx[9]
By the sum rule, differentiate -3x²+2?
d/dx[-3x²]+d/dx[2]
Find the derivative of cos(lnx) using the chain rule
d/dx[f(g(x))]=f'(g(x))g'(x), where f(x)=cosx and g(x)= lnx =-sin(ln(x))/x
find derivative of f(x)= x²/ln(2)?
differentiate using the power rule, d/dx[x^n]=nx^n-1 2x/ln(2)
y=x²+4x at x=-5 is -5, true or false?
false. 2x+4...2(-5)+4=-6
find derivative of f(x)= √7?
0
derivative of line "x" or "ax"?
1, a
find derivative of f(x)= x-x³?
1-3x²
Negative Exponent Rule
x ⁻ⁿ = 1/xⁿ
find derivative of f(x)= x²-2?
2x
find derivative of f(x)= x(x+1)?
2x+1
find derivative of f(x)= x²-1/x?
2x+1/x²
find derivative of f(x)= x²+3x-6?
2x+3
differentiate each function with respect to X. Problems may contain constants a, b, and c, y=4ax^3a-bx^3c...?
4a(3a)x^(3a-1)-3cbx^3c-1
find derivative of f(x)= 2x²-4?
4x
find derivative of f(x)= 2x²-5x+10?
4x-5
cos(9x)[sin(9x)]^-1, 5th & 6th step of derivative...?
5. Raise cos(9x) to the power of 1. -->(cos'(9x)cos(9x))sin^-2(9x)d/dx[9x]+sin^-1(9x)d/dx[cos(9x)] 6. Use the power rule a^m×a^n=a^(m+n) to combine exponents -cos(9x)^1+1sin^-2(9x)d/dx[9x]
find derivative of f(x)= x²/2-3x?
x-3
derivative of a^x, exponential...?
xa^x-1
derivative of √x?
½x^-½
find derivative of f(x)= ⅓x?
⅓
find derivative of f(x)= x√3?
√3 √3×d/dx×x
find derivative of f(x)= √(3x)?
√3/2√x
Constant Multiple Rule
(cf)'=cf', The derivative of a constant times a function = the constant times the derivative of a function.
derivative quotient rule, f/g...?
(f'g-g'f)/g²
y=e^(-x+2), derivative...?
-1/e^2
find derivative of f(x)= 1/ x¹⁰⁰⁰?
-1000x^-1001
find derivative of f(x)= 2/x?
-2x^-2
find derivative of f(x)= 8/√x-3x?
-4x^-3/2-3
a classmate claims. F x g equals F' X g' for any functions f and g. Show an example that proves your classmate is wrong
8≠0 1. Set up the composite result function f(g(x)) 2. Evaluate f(g(x)) by substituting the value of g into f 3. Multiply 2 * 4
derivative reciprocal rule, 1/f=f^-1...?
-f'/f²
derivative of cos(x)?
-sinx
f(x)=-2x^¼, derivative...?
-½x^-¾
derivative of constant "c"?
0
differentiate each function with respect to X. Problems may contain constants a, b, and c, y=5c
0
find derivative of f(x)= 1-5?
0
find derivative of f(x)= 5²?
0
cos(9x)[sin(9x)]^-1, 7th step of derivative...?
1. Add 1 and 1 cos²(9x)sin^-2(9x)d/dx[9x]+sin^-1(9x)d/dx[cos(9x)] 2. Since 9 is constant with respect to x, the derivative of 9x with respect to x is 9 d/dx[x] 3. cos²(9x)sin^-2(9x)(9d/dx[x])+sin^-1(9x)d/dx[cos(9x)] 4. Multiply 9 by -1 9cos²(9x)sin^-2+sin^-1(9x)d/dx[cos(9x)]
Diagnostic #4 - health club Offers membership and a rate of $300 for the 1st 50 people who join. For each member in excess of 50 the membership fee will be reduced by $2 for all members. Due to space limitations, at most a 125 memberships will be sold. How many memberships should the club sell in order to maximize its revenue?
1. Current rate: $300 # of members: 50 Price=> P= 300-2(x-50) 2. P= 300-2x+100=> P=400-2x where 50≤A≤125 3. •Revenue = price × quantity •R(x)= (400-2x)×x=>400-2x² 4. To maximize, put R'(x)=0, R'(x)= 400-4x=0 where x=100, average 100 Competency 12.4- Calculate derivatives of functions (for example: polynomial, exponential)...
y=log₃(3x⁵+5)⁵, 1-4 differentiation steps?
1. Differentiate using chain rule. 2. Apply the chain rule, set u = 3x⁵+5, f(x)=log₃(u)⁵ 3. 1/uln(3)×d/dx(3x⁵+5) 4. Replace all occurrences of "u" with 3x⁵+5
differentiate y=e^-x+2 at x=4..steps...?
1. Differentiate using the chain rule where f(x)=e^x and g(x)=-x+2 2. Differentiate g(x) using the sum rule. 3. Differentiate using the power rule. 4. Simplify the expression, -e^-x+2 5. Evaluate the derivative ar x=4, -1/e²
cos(9x)[sin(9x)]^-1, 1st step of derivative...?
1. Differentiate using the chain rule which states that d/dx[f(g(x)] is f'(g(x))g'(x) where f(x)=x^-1 and g(x)=sin(9x) 2. To apply the chain rule, set u as sin(9x) cos(9x)(d/du[u₁^-1]d/dx[sin(9x)])+sin^-1(9x)d/dx[cos(9x)]
cos(9x)[sin(9x)]^-1, 4th step of derivative...?
1. Differentiate using the chain rule, which states that d/dx [f(g(x)] is f'(g(x))g'(x) where f(x)=sin(x) and g(x)=9x 2. To apply the chain rule, set u₂ as 9x. cos(9x)sin^-2(9x)(d/du₂[sin(u₂)×d/dx[9x])+sin^-1(9x)d/dx[cos(9x)]. --> The derivative of sin(u₂) with respect to u₂ is cos(u₂)
cos(9x)[sin(9x)]^-1, 8th step of derivative...?
1. Differentiate using the chain rule, which states that d/dx[f(g(x)] is f'(g(x))g'(x) where f(x)=cos(x) and g(x)=9x 2. To apply the chain rule, set u₃ as 9x 9cos²(9x)sin^-2(9x)+sin^-1(9x)(d/du₃[cos(u₃)]d/dx[9x] 3. The derivative of cos(u₃) with respect to u₃ is -sin(u₃), 9cos²(9x)sin^-2(9x)+sin^-1(9x)(-sin(u₃)d/dx[9x] 4. Replace all occurrences of u₃ with 9x, 9cos²(9x)sin^-2(9x)+sin^-1(9x)(-sin(9x)d/dx[9x]
differentiate cos²(x³)=
1. Differentiate using the chain rule. f(g(x)) where f(x)= x² and g(x)=cos(x³) 2. 2cos(x³)d/dx[cos(x³)] 3. Differentiate using the power rule. 2cos(x³)-sin(x³)= -6x²cos(x³)sin(x³)
cos(9x)[sin(9x)]^-1, 2nd step a) of derivative...?
1. Differentiate using the power rule which states that d/du₁[u₁^n] is nu₂^n-1 where n=-1 cos(9x)×(-u₁^-2)d/dx 2. To apply the chain rule, set u as sin(9x) cos(9x)(d/du[u₁^-1]d/dx[sin(9x)])+sin^-1(9x)d/dx[cos(9x)]
Differentiate y=x+1/x²-1
1. d/dx (x+1)/(x²-1) 2. Apply the Quotient Rule: (f/g)'= [f×'g - g'×f]/g² 3. =[d/dx(x+1)(x²-1)-d/dx(x²-1)(x+1)]/(x²-1)² 4. d/dx(x+1)=1 5. d/dx(x²-1)=2x 6. =[1×(x²-1)-2x(x+1)]/(x²-1)²=-1/(x-1)² Competency 12.4 calculate the derivatives of functions (polynomial)
Differentiate f(x)=tan(cosx)
1. differentiate using the chain rule Which States that d/dx[f(g(x))] is f'(g(x))g'(x) where f(x)=tan(x) and g(x)= cos(x) 2. To apply the chain rule, set u as cos(x)... d/du[tan(u)]×d/dx[cos(x)] 3. The derivative of tan(u) with respect to u is sec²(u), sec²(u)d/dx[cos(x)] 4. Replace all occurrences of u with cos(x), sec²(cos(x))d/dx[cos(x)] 5. The derivative of cos(x) with respect to x is -sinx, ∴sec²(cos(x))(-sin(x)) 6. Move -1 to the left of sec²(cos(x)), -1×sec²(cos(x))sin(x) 7. Rewrite -1sec²(cos(x)) as -- sec²(cos(x)), -sec²(cos(x))sin(x) Competency 12.4- Calculate the derivatives of functions (for example: polynomial, exponential logarithmic)...
derivative of tan^-1(x)?
1/(1 + x^2), rate of change of arccos with respect to x.
derivative of log₃(u)...?
1/uln(3)
derivative of logarithm lnx?
1/x
derivative of cos^-1(x)?
1/√(1-x^2)
derivative of sin^-1(x)?
1/√(1-x^2)
sin^-1(x)
1/√(1-x^2)
find derivative of f(x)= x¹⁰⁰⁰?
1000x⁹⁹⁹
find derivative of f(x)= 4x^5/2?
10x^3/2
find the first derivative of the function: f(x)= x³-6x²+5x+4
12.4 - Calculate the derivatives of functions (e.G. polynomial, exponential, logarithmic) 3x²-12x+5 use the power rule for polynomial differentiation: if Y equals AX^n, then Y'=nax^n-1.
find derivative of f(x)= 2x^10-4x²?
20x⁹-8x
f(x)=2sin(2x) at x=-π/2, differentiate using the chain rule where f(x)=sin(x) and g(x)= 2x..?
2[cos(2x)d/dx(2x)]
derivative of "x²"?
2x
#84- Which of the following is an equation of the line tangent to the curve y= x²- 4X at the point (1, -3)?
Competency 12.4 this question requires examinee to analyze the derivative as the slope of a tangent line and as a limit of the difference quotient. The slope of the line tangent to y= x²-4x for any x is given by y'= 2x-4 for x=1, y'=2(1)-4=2, so the slope is -2. The equation of the line that goes through the point (1,-3) with a slope -2 is y=-2x-1.
#90- Which of the following equations represents the derivative of the function f(x)=4√x?
Competency 12.4- this question requires examinee to calculate the derivatives of functions. For f(x)=4√x=4(x)½ use the power rule: f'(x)=½4(x)^-1/2= 2/√x=2√x/x
find derivative of f(x)= 6x+5-(3x+3x²)?
Differentiate using the power rule, d/dx[x^n]=nx^n-1 -6x+3
y=(-2x⁴-3)(-2x²+1), steps 1 -3, true or false...?
False, #1, -2x²+1...? 1. Take the derivative using the product rule, f(x)=-2x⁴-3, g(x)= -2x²+2 2. By the sum rule, x is (d/dx[-2x²]+d/dx[1] 3. Evaluate d/dx[-2x²] (-2x⁴-3)(-4x+d/dx[1])+(-2x²+1)d/dx[-2x⁴-3]
y=(4^x^3+2)³, differentiation steps - true or False...?
False. 1. Differentiate using chain rule, f(x)=x² and g(x)=4^x^3+2 2. By the sum rule, (d/dx[4^x^3]+d/dx[2]) 3. Differentiate using chain rule where f(x)=4^x and g(x)=x³ 4. Replace all occurrences of "u" with 3x⁵+5
y=5x²+3, by the sum rule the derivative with respect to x is d/dx[-5x²]+[3], true or false...?
False. d/dx[-5x²]+d/dx[3], true or false...?
Find the derivative. 11. y = (a+b/x^2)^3
Given a function, y = (a+b/x^2)^3, we need to find the derivative Step 1 - dy/dx = 3(a+b/x^2)^2(b(-2/x^3)) =-6b/x^2(a+b/x^2)^2 Therefore, dy/dx = 6b/x^3(a+b/x^2)^2
Find the derivative. f(ϴ) = cot(3ϴ + π)
Given the function f(ϴ) = cot(3ϴ+ π) Step 1 - We need to find the derivative. F'(ϴ) = -(csc^2(3ϴ + π))(3) =-3csc^2(3ϴ + π) Therefore, f'(ϴ) = -3csc^2(3ϴ+ π)
cos(9x)[sin(9x)]^-1, 3rd step of derivative...?
Simplify the expression 1. move -1 to the left of cos( 9x) -1×cos(9x)sin^-2(9x)d/dx[sin(9x)]+sin^-1(9x)d/dx[cos(9x)] 2. Rewrite -1 cos(9x) as -cos(9x) -cos(9x)×sin^-2(9x)d/dx[sin(9x)]+sin^-1(9x)d/dx[cos(9x)]
find derivative of f(x)= x²-e²?
Since -e² is constant with respect to x, the derivative of -e² with respect to x is 0.
Find the second derivative F(x) = cos(2-x)
Step 1 - We need to find the derivative. F'(x) = -(sin(2-x))(-1) =sin(2-x) F'(x)=cos(2-x)
y=4^(4x^4), apply chain rule...set u as 4x⁴?
Step 1- Differentiate using the chain rule, which states that d/dx[f(g(x))] is f'(g(x))g'(x) where f(x)=4x and g(x)=4x⁴ a) to apply the chain rule, set u as 4x⁴, d/du[4⁴]d/dx b) Differentiate using the exponential role with states that d/du[a^u] is a^uln(a) where a =4 4^uln(u)d/dx[4x⁴] c) Replace all occurrences of u with 4x⁴ 4^(4x⁴)ln(4)d/dx[4x⁴] Step 2 - since 4 is constant with respect to x, the derivative of 4x⁴ with respect to x is 4d/dx[x⁴] Step 3 - raise 4 to the power of 1, 4¹*4^4x⁴ln(4)d/dx[x⁴] Step 4 - Use the power rule a^m*a^n=a^m+n to combine exponents, 4^1+4x⁴ln(4)d/dx[x⁴] Step 5 - Differentiate using the power rule which states that d/dx[x^n] is nx^n-1 where n=4 4^1+4x⁴ln(4)(4x³) Step 6- Raise 4 to the power of 1, 4¹*4^1+4x⁴ln(4)x³ Step 7 - use the power rule, a^m+a^n=a^m+n combine exponents, 4^1+1+4x⁴ln(4)x³ Step 8 - add 1 and 1, 4^(2+4x⁴)ln(4)x³ Step 9 - reorder the factors of 4^2+4x¹(x³ln(4))
derivative of logₐ(x)?
Step 1- rewrite logₐ(x) using the change of base formula. a) the change of base rule can be used if a and b are greater than zero. d/da[logₐ(x)=logᵦ(x)/logᵦ(a)] b) substitute in values for the variables in the change of base formula, using b=e d/da[ln(x)/ln(a)] Step 2- Differentiate using the constant multiple rule. a)since ln(x) is constant with respect to a, the derivative of ln(x)/ln(a) with respect to a is ln(x)d/da[1/ln(x)] b)Rewrite 1/ln(a) as ln^-1(a), ∴ ln(x)d/da[ln^-1(a)] Step 3- differentiate using the chain rule, which states that d/da[f(g(a))] is f'(g(a))g'(a) where f(a)=a^-1 and g(a)=ln(a) Step 4- the derivative of ln(a) with respect to a is 1a. ln(x)(-ln^-2(a)1/a) Step 5- combine fractions, -ln(x)/aln²(a)
f(x)=-tan(2x)
Step 1- since -1 is constant with respect to x, the derivative of -tan(2x) with respect to x is d/dx[tan(2x)] Step 2- Differentiate using the chain rule, which states d/dx[f(g(x))] is f'(g(x))g'(x) where f(x)=tan(x) and g(x)=2x Step 3- -(sec²(2x))d/dx[2x] Step4-=-2sec²(2x) <)=> -[sec²(2x)d/dx(2x)]
find derivative of f(x)= x-1/x?
Step 1: 1+ d/dx[-1/x] Step 2: d/dx[-1/x]=1+1/x² Step 3: Rewrite the expression using the negative exponent rule and reorder terms b^-n=1/b^n, 1+x^-2
find derivative of f(x)= 5x³-5?
Step 1: By the sum rule, the derivative of 5x³-5 with respect to x is d/dx[5x³]+d/dx[-5]. Srep2: Differentiate using the constant rule, 15x².
y=log₅(-5x³-2)³, differentiate...?
Step 1: Differentiate using the chain rule, which states that d/dx[f(g(x))]*g'(x) where f(x) = log₅(x) and g(x)= (-5x³-2)³, 1/(-5x³-2)³ln(5)d/dx[(-5x³-2)³] Step2: Differentiate using the chain rule which states d/dx[f(g(x))] is f'(g(x))g'(x) f(x)=x³ and g(x)=-5x³-2, 1/(-5x³-2)³ln5(3(-5x³-2)²d/dx[-5x³-2]) -45x²/-5x³ln(5)-2ln(5)
find derivative of f(x)= x⁴/4+x-2?
Step 1: Subtract 2 from 4, d/dx[x⁴/x+2] Step 2: Differentiate using the quotient rule, d/dx[f(x)/g(x)] is g(x)d/dx[f(x)]-f(x)d/dx[g(x)]/g(x)²; f(x) = x⁴ and g(x) = x+2 Step 3: Differentiate. 4(x+2)x³-x⁴/(x+2)² Step 4: x³(3x+8)/(x+2)²
Find the derivative of e^ 5x²+7x-13 using the power rule
Step 1: by the sum rule, the derivative of e^5x²+7x-13 with respect to x is d/dx[e^5x²]+d/dx[7x]+d/dx[-13] Step 2: Evaluate d/dx[e^5x²] a) Differentiate using the chain rule, which states that d/dx[f(g(x))] is f'(g(x)g'(x)) is: (i) to apply the chain rule, set u as 5x² d/dx[e^u]d/dx[5x²]+d/dx[7x]+d/dx[-13] (ii) Replace all occurrences of u with 5x² (iii) since 5 is constant with respect to x, the derivative of 5x² with respect to x is 5d/dx[x²]e^5x²(5d/dx[x²]+d/dx[7x]+d/dx[-13] (iv) Differentiate using the power rule which states that d/dx[x^n] is nx^n-1 where n=2, e^5x²(5(2x))+d/dx[x²]+d/dx[7x]+d/dx[-13] (v) Multiply 2 by 5. e^5x²(10x)+d/dx[7x]+d/dx[-13] Step 3: Evaluate d/dx[7x] e^5x²(10x)+7+d/dx[-13] Step 4: since -13 is constant with respect to x, the derivative of -13 with respect to x is u. e^5x²(10x)+7+0 (i) add e^5x²(10x)+7 (ii) Reorder items, 10e^5x²(x+7) (iii) Reorder factors in above step, 10xe^5x²+7
find derivative of f(x)= 3√x?
Step 1: use ^n√a^x = a^x/n to rewrite √x as x^½, now d/dx [3x^½] Step 2: Since 3 is constant with respect to x, the derivative of 3x^½ with respect to x Step 3: where n = ½, 3(½x^½-1) Step 10: Combine 3 and x^-½/2, 3x^-½/2 Step 11: move x^-½ to the denominator using the negative exponent rule. b^-n= 1/b^n 3/2x^-½
f(x)=(-3x+9)^½, differentiate using power rule?
Yes, you can... Step 1: Differentiate using the chain rule, f'(g(x))*g'(x), f(x)=x^½, g(x)=-3x+9. Let u= -3x+9 (x^½)d/dx*(-3x+9)d/dx ½(3x-9)*d/dx[3x+9] Step 2: by the sum rule, d/dx[3x]+d/dx[9] -3/2(-3x+9)^½
competency 12.4 - calculate the derivatives of functions polynomial exponential and logarithmic
ability to calculate the first and second derivatives on a variety of functions is required.
Chain Rule (using d/dx)
dy/dx = dy/du * du/dx
derivative of e^x, exponential...?
e^x
Chain Rule f(g(x))
f'(g(x)) x g'(x)
derivative sum rule, f+g...?
f'+g'
derivative difference rule, f-g...?
f'-g'
f(x)=(-7x⁴+10x^⅖+8)(x²+10), apply product rule?
f(x)=-7x⁴+10x^⅖+8 g(x)=x²+10
derivative product rule, fg...?
fg'+gf'
derivative of tan(x)?
sec²(x)
apply chain rule, y=log₂(4x²)...?
set f(x)= log₂(x) and g(x)= 4x², set u = 4x²
differentiate, bring constant, "3" out front, true or false..., 3d/dx[x²]?
true
y=3^(cos3x⁴) step 4: replace all occurrences of "U₂" with 3x⁴. Step 5, differentiate with constant multiple rule?
true -(3×3^(cos(3x⁴))ln(3)sin(3x⁴)d/dx[x⁴]
y=(-6x⁴+2+6x^-4)(6x⁴+7), differentiate doing the following steps, true or false?
true 1. Take the derivative using the power rule 2. Use the sum rule to differentiate d/dx[6x⁴]d/dx[7] 3. Evaluate, d/dx [6x⁴] 4. Differentiate 5. Evaluate d/dx[-6x⁴]
f(x)=(5x⁵+5)(-2x⁵-3), step 1, differentiate using the product rule, true or false...?
true, product rule
f(x)=(5+3x^-2)(4x⁵+6x³+10), differentiate using the product rule, f(x)=5+3x^-2 and g(x)= 4x⁵+6x³+10, true or false...?
true. (5+3x^-2)d/dx[4x⁵+6x³+10]+(4x⁵+6x³+10)d/dx[5+3x^-2]
y=(-x+4)^½, differentiation steps - true or False...?
true. 1. Differentiate using chain rule, f(x)=x½ and g(x)=-x+4 2. By the constant multiple rule, move -1 out in front of the expression, ½(-x+4)^-½×d/dx[x-4] 3. Rewrite the expression using the negative exponent rule = 1/b^n = b^-n 4. By the sum rule, -1/{2(-0+4)^½} =-¼
y=-ln(x+3), differentiation steps - true or False...?
true. 1. Since -1 is a constant with respect to x, the derivative of -ln(x+3) with respect to x is -d/dx[ln(x+3)]. 2. Differentiate using the chain rule, f(x)= lnx and g(x)=x+3 3. Set u= x+3, -(d/du×[ln(u)]×d/dx[x+3]) 4. The derivative of ln(u) is 1/u
y=(x⁴+3)(-4x⁵+5x⁴+5), steps to differentiate...true or false?
true. 1. Take the derivative using the product rule 2. Use the sum rule 3. Evaluate and differentiate
y=(-10x²-7x^⅖+9)(2x³+4), steps to differentiate...true or false?
true. 1. Take the derivative using the product rule 2. Use the sum rule to differentiate d/dx[2x³]d/dx[4] 3. Evaluate, d/dx [-10x²], d/dx[-7x^⅖] and differentiate
f(x)=x²(-3x²-2), steps...true or false?
true. 1. Take the derivative using the product rule, f(x)=x², g(x)= -3x²-2 2. By the sum rule, x²(d/dx[-3x²]+d/dx[-2]
y=log₅tan4x⁴, step4: (1/tan4x⁴)×ln5×d/dx[tan4x⁴], step 5, dy/dx (16x³×sec²4x⁴)/tan4x⁴×ln5..true or false?
true. dy/dx= 1/tan4x⁴×1/ln5×sec²4x⁴×16x³
y=2sin(2x) at x=-π/2, step1 - true or False...?
true. 1. Since 2 is constant with respect to x, the derivative of 2sin(x) with respect to x is: 2d/dx[sin(2x)]
d/dx e^(-x+2) is e^(-x+2), true or false...?
true. e^(-x+2)
