Derivative of y = ln ⁡ x y=\ln x y = ln x

y ′ = 1 x y'=\frac{1}{x} y ′ = x 1

y ′ = ln ⁡ x y'=\ln x y ′ = ln x

y ′ = 1 ln ⁡ x y'=\frac{1}{\ln x} y ′ = ln x 1

Derivative of y = e 5 x y=e^{5x} y = e 5 x

y ′ = 5 e 5 x y'=5e^{5x} y ′ = 5 e 5 x

y ′ = 5 e 4 x y'=5e^{4x} y ′ = 5 e 4 x

y ′ = e 5 x y'=e^{5x} y ′ = e 5 x

y ′ = 5 e 5 x − 1 y'=5e^{5x-1} y ′ = 5 e 5 x − 1

Derivative of y = 5 x y=5^x y = 5 x

y ′ = 5 x ⋅ ln ⁡ 5 y'=5^x\cdot\ln5 y ′ = 5 x ⋅ ln 5

y ′ = x 5 x − 1 y'=x5^{x-1} y ′ = x 5 x − 1

y ′ = 5 x ln ⁡ 5 y'=\frac{5^x}{\ln5} y ′ = ln 5 5 x

y ′ = e x ⋅ ln ⁡ 5 y'=e^x\cdot\ln5 y ′ = e x ⋅ ln 5

Fundamental Theorem: ∫ a b f ′ ( x ) d x = \int_a^bf'\left(x\right)dx= ∫ a b f ′ ( x ) d x =

f ( b ) − f ( a ) f\left(b\right)-f\left(a\right) f ( b ) − f ( a )

f ( a ) − f ( b ) f\left(a\right)-f\left(b\right) f ( a ) − f ( b )

f ′ ( b ) − f ′ ( a ) f'\left(b\right)-f'\left(a\right) f ′ ( b ) − f ′ ( a )

f ′ ( a ) − f ′ ( b ) f'\left(a\right)-f'\left(b\right) f ′ ( a ) − f ′ ( b )

Fundamental Theorem: f ( b ) = f\left(b\right)= f ( b ) =

f ( a ) + ∫ a b f ′ ( x ) d x f\left(a\right)+\int_a^bf'\left(x\right)dx f ( a ) + ∫ a b f ′ ( x ) d x

f ( a ) − ∫ a b f ′ ( x ) d x f\left(a\right)-\int_a^bf'\left(x\right)dx f ( a ) − ∫ a b f ′ ( x ) d x

f ( a ) + ∫ a b f ( x ) d x f\left(a\right)+\int_a^bf\left(x\right)dx f ( a ) + ∫ a b f ( x ) d x

f ( a ) − ∫ a b f ( x ) d x f\left(a\right)-\int_a^bf\left(x\right)dx f ( a ) − ∫ a b f ( x ) d x

Derivative of f ( x ) = ∫ 4 x sin ⁡ ( t ) d t f\left(x\right)=\int_4^x\sin\left(t\right)dt f ( x ) = ∫ 4 x sin ( t ) d t

f ′ ( x ) = sin ⁡ ( x ) f'\left(x\right)=\sin\left(x\right) f ′ ( x ) = sin ( x )

f ′ ( x ) = cos ⁡ ( x ) f'\left(x\right)=\cos\left(x\right) f ′ ( x ) = cos ( x )

f ′ ( x ) = sin ⁡ x − sin ⁡ 4 f'\left(x\right)=\sin x-\sin4 f ′ ( x ) = sin x − sin 4

f ′ ( x ) = − cos ⁡ x + cos ⁡ 4 f'\left(x\right)=-\cos x+\cos4 f ′ ( x ) = − cos x + cos 4

A = 1 2 h ( b 1 + b 2 ) A=\frac{1}{2}h\left(b_1+b_2\right) A = 2 1 h ( b 1 + b 2 )

A = 2 l + 2 w A=2l+2w A = 2 l + 2 w

A = 1 4 h ( b 1 + b 2 ) A=\frac{1}{4}h\left(b_1+b_2\right) A = 4 1 h ( b 1 + b 2 )

A = 1 2 b h A=\frac{1}{2}bh A = 2 1 b h

Average rate of change formula

f ( b ) − f ( a ) b − a \frac{f\left(b\right)-f\left(a\right)}{b-a} b − a f ( b ) − f ( a )

1 b − a ∫ a b f ( x ) d x \frac{1}{b-a}\int_a^bf\left(x\right)dx b − a 1 ∫ a b f ( x ) d x

f ( b ) − f ( a ) 2 \frac{f\left(b\right)-f\left(a\right)}{2} 2 f ( b ) − f ( a )

Average value of a function

1 b − a ∫ a b f ( x ) d x \frac{1}{b-a}\int_a^bf\left(x\right)dx b − a 1 ∫ a b f ( x ) d x

f ( b ) − f ( a ) b − a \frac{f\left(b\right)-f\left(a\right)}{b-a} b − a f ( b ) − f ( a )

f ( b ) − f ( a ) 2 \frac{f\left(b\right)-f\left(a\right)}{2} 2 f ( b ) − f ( a )

final position - initial position change in time \frac{\text{final position - initial position}}{\text{change in time}} change in time final position - initial position

final velocity - initial velocity change in time \frac{\text{final velocity - initial velocity}}{\text{change in time}} change in time final velocity - initial velocity

final velocity - initial velocity 2 \frac{\text{final velocity - initial velocity}}{2} 2 final velocity - initial velocity

∫ e x d x = \int e^x\ dx= ∫ e x d x =

ln ⁡ ( x ) + C \ln\left(x\right)+C ln ( x ) + C

1 2 e 2 x + C \frac{1}{2}e^{2x}+C 2 1 e 2 x + C

1 x + 1 e x + 1 + C \frac{1}{x+1}e_{ }^{x+1}\ +C x + 1 1 e x + 1 + C

∫ e 4 x d x = \int e^{4x}\ dx= ∫ e 4 x d x =

1 4 e 4 x + C \frac{1}{4}e^{4x}+C 4 1 e 4 x + C

4 e 4 x + C 4e^{4x}+C 4 e 4 x + C

4 e 3 x + C 4e^{3x}+C 4 e 3 x + C

1 5 e 5 x + C \frac{1}{5}e^{5x}+C 5 1 e 5 x + C

What equation do we use to find the equation of a tangent line?

y − y 1 = m ( x − x 1 ) y-y_1=m\left(x-x_1\right) y − y 1 = m ( x − x 1 )

x − x 1 = m ( y − y 1 ) x-x_1=m\left(y-y_1\right) x − x 1 = m ( y − y 1 )

y + y 1 = m ( x + x 1 ) y+y_1=m\left(x+x_1\right) y + y 1 = m ( x + x 1 )

x + x 2 = m ( y + y 1 ) x+x_2=m\left(y+y_1\right) x + x 2 = m ( y + y 1 )

AP Calculus AB Full Year Review

Authored by Jorge Ibanez

Mathematics

11th Grade - University

CCSS covered

Used 64+ times

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MULTIPLE CHOICE QUESTION

5 sec • 1 pt

$y'=5e^{5x}$

$y'=5e^{4x}$

$y'=e^{5x}$

$y'=5e^{5x-1}$

Derivative of $y=5^x$

$y'=x5^{x-1}$

$y'=5^x\cdot\ln5$

$y'=\frac{5^x}{\ln5}$

$y'=e^x\cdot\ln5$

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AP Calculus AB Full Year Review

Derivative of y=x7y=x^7y=x7

Derivative of y=sin⁡xy=\sin xy=sinx

Derivative of y=cos⁡xy=\cos xy=cosx

Derivative of y=tan⁡xy=\tan xy=tanx

Derivative of y=cot⁡xy=\cot xy=cotx

Derivative of y=sec⁡xy=\sec xy=secx

Derivative of y=csc⁡xy=\csc xy=cscx

Derivative of y=ln⁡xy=\ln xy=lnx

Derivative of y=e5xy=e^{5x}y=e5x

Derivative of y=5xy=5^xy=5x

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