∫ ( a x ) d x = \int_{ }^{ }\left(a^x\right)dx= ∫ ( a x ) d x =

a x log ⁡ a + C \frac{a^x}{\log a}+C lo g a a x + C

a x log ⁡ a + C a^x\log a+C a x lo g a + C

∫ d x = \int dx= ∫ d x =

u n d e f i n e d undefined u n d e f i n e d

If g ( x ) g\left(x\right) g ( x ) and f ( x ) f\left(x\right) f ( x ) are inverse of each other, then

g ′ ( x ) = 1 f ′ ( g ( x ) ) g'\left(x\right)=\frac{1}{f'\left(g\left(x\right)\right)} g ′ ( x ) = f ′ ( g ( x ) ) 1

g ′ ( x ) = 1 f ′ ( x ) g'\left(x\right)=\frac{1}{f'\left(x\right)} g ′ ( x ) = f ′ ( x ) 1

g ′ ( x ) = f ( g ( x ) ) g'\left(x\right)=f\left(g\left(x\right)\right) g ′ ( x ) = f ( g ( x ) )

g ′ ( x ) = g ( f ( x ) ) g'\left(x\right)=g\left(f\left(x\right)\right) g ′ ( x ) = g ( f ( x ) )

The derivative of the function, f(x) = aln(x)

f ′ ( x ) = a x f'\left(x\right)=\frac{a}{x} f ′ ( x ) = x a

f ′ ( x ) = a ln ⁡ ( x ) f'\left(x\right)=a\ln\left(x\right) f ′ ( x ) = a ln ( x )

f ′ ( x ) = a e x f'\left(x\right)\ =ae^x f ′ ( x ) = a e x

CHECK ALL THAT APPLY What are all the ways to signify a derivative

lim ⁡ h → 0 f ( x + h ) − f ( x ) h \lim_{h\rightarrow0}\frac{f\left(x+h\right)-f\left(x\right)}{h} h → 0 lim h f ( x + h ) − f ( x )

f ′ ( x ) f'\left(x\right) f ′ ( x )

d d x [ f ( x ) g ( x ) ] \frac{\text{d}}{\text{d}x}\left[f\left(x\right)g\left(x\right)\right] d x d [ f ( x ) g ( x ) ]

f ′ ( x ) g ( x ) + g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) + g ′ ( x ) f ( x )

f ′ ( x ) g ′ ( x ) f'\left(x\right)g'\left(x\right) f ′ ( x ) g ′ ( x )

f ′ ( x ) g ( x ) g ′ ( x ) f'\left(x\right)g\left(x\right)g'\left(x\right) f ′ ( x ) g ( x ) g ′ ( x )

f ′ ( x ) g ( x ) − g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)-g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) − g ′ ( x ) f ( x )

d d x ( x n ) = \frac{\text{d}}{\text{d}x}\left(x^n\right)= d x d ( x n ) =

n x ( n − 1 ) nx^{\left(n-1\right)} n x ( n − 1 )

d d x [ f ( x ) + g ( x ) ] \frac{\text{d}}{\text{d}x}\left[f\left(x\right)+g\left(x\right)\right] d x d [ f ( x ) + g ( x ) ]

f ′ ( x ) + g ′ ( x ) f'\left(x\right)+g'\left(x\right) f ′ ( x ) + g ′ ( x )

f ′ ( x ) g ( x ) + g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) + g ′ ( x ) f ( x )

f ′ ( x ) g ( x ) g ′ ( x ) f'\left(x\right)g\left(x\right)g'\left(x\right) f ′ ( x ) g ( x ) g ′ ( x )

f ′ ( x ) g ( x ) − g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)-g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) − g ′ ( x ) f ( x )

d d x [ f ( x ) − g ( x ) ] \frac{\text{d}}{\text{d}x}\left[f\left(x\right)-g\left(x\right)\right] d x d [ f ( x ) − g ( x ) ]

f ′ ( x ) − g ′ ( x ) f'\left(x\right)-g'\left(x\right) f ′ ( x ) − g ′ ( x )

f ′ ( x ) g ( x ) + g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) + g ′ ( x ) f ( x )

− f ′ ( x ) g ( x ) g ′ ( x ) -f'\left(x\right)g\left(x\right)g'\left(x\right) − f ′ ( x ) g ( x ) g ′ ( x )

f ′ ( x ) g ( x ) − g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)-g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) − g ′ ( x ) f ( x )

If c is a constant, d d x [ c ⋅ f ( x ) ] \frac{\text{d}}{\text{d}x}\left[c\cdot f\left(x\right)\right] d x d [ c ⋅ f ( x ) ]

c ⋅ f ′ ( x ) c\cdot f'\left(x\right) c ⋅ f ′ ( x )

c ′ ⋅ f ′ ( x ) c'\cdot f'\left(x\right) c ′ ⋅ f ′ ( x )

c ′ ⋅ f ( x ) ⋅ f ′ ( x ) c'\cdot f\left(x\right)\cdot f'\left(x\right) c ′ ⋅ f ( x ) ⋅ f ′ ( x )

d d x [ f ( g ( x ) ) ] = \frac{\text{d}}{\text{d}x}\left[f\left(g\left(x\right)\right)\right]= d x d [ f ( g ( x ) ) ] =

f ′ ( g ( x ) ) g ′ ( x ) f'\left(g\left(x\right)\right)g'\left(x\right) f ′ ( g ( x ) ) g ′ ( x )

f ′ ( g ′ ( x ) ) f'\left(g'\left(x\right)\right) f ′ ( g ′ ( x ) )

f ′ ( x ) g ′ ( x ) f'\left(x\right)g'\left(x\right) f ′ ( x ) g ′ ( x )

f ′ ( x ) g ( x ) + g ′ ( x ) f ( x ) f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right) f ′ ( x ) g ( x ) + g ′ ( x ) f ( x )

d d x ( sin ⁡ x ) = \frac{\text{d}}{\text{d}x}\left(\sin x\right)= d x d ( sin x ) =

− sin ⁡ x -\sin x − sin x

− cos ⁡ x -\cos x − cos x

d d x ( cos ⁡ x ) = \frac{\text{d}}{\text{d}x}\left(\cos x\right)= d x d ( cos x ) =

− sin ⁡ x -\sin x − sin x

− cos ⁡ x -\cos x − cos x

d d x ( ln ⁡ x ) = \frac{\text{d}}{\text{d}x}\left(\ln x\right)= d x d ( ln x ) =

1 x ln ⁡ x \frac{1}{x\ln x} x ln x 1

d d x ( log ⁡ a x ) = \frac{\text{d}}{\text{d}x}\left(\log_ax\right)= d x d ( lo g a x ) =

1 x ln ⁡ a \frac{1}{x\ln a} x ln a 1

1 x ln ⁡ x \frac{1}{x\ln x} x ln x 1

ln ⁡ a x \frac{\ln a}{x} x ln a

d d x ( a x ) = \frac{\text{d}}{\text{d}x}\left(a^x\right)= d x d ( a x ) =

1 x ln ⁡ a \frac{1}{x\ln a} x ln a 1

d d x ( e x ) = \frac{\text{d}}{\text{d}x}\left(e^x\right)= d x d ( e x ) =

1 x ln ⁡ e \frac{1}{x\ln e} x ln e 1

Derivative and Integration Rules

Authored by Brianna Considine

Mathematics

11th Grade

CCSS covered

Used 11+ times

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MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Determine the integral rule of the given expression:

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Determine the integral rule of the given expression:

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Determine the integral rule of the given expression:

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Determine the integral rule of the given expression:

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Determine the integral rule of the given expression:

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Determine the integral rule of the given expression:

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Determine the integral rule of the given expression:

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