Calc 1 Formulas to Memorize

Derivative of a constant

0

Derivative of x

1

Constant multiple rule

d/dx[c f(x)] = c f'(x)

Sum rule

d/dx[f(x) + g(x)] = f'(x) + g'(x)

Difference rule

d/dx[f(x) - g(x)] = f'(x) - g'(x)

Power rule

d/dx[x^n] = n x^(n-1)

Product rule

d/dx[fg] = f'g + fg'

Quotient rule

d/dx[f/g] = (f'g - fg')/g^2

Chain rule

d/dx[f(g(x))] = f'(g(x))g'(x)

Derivative of sqrt(x)

1/(2sqrt(x))

Derivative of 1/x

-1/x^2

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec^2(x)

Derivative of cot(x)

-csc^2(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of sin(u)

cos(u)u'

Derivative of cos(u)

-sin(u)u'

Derivative of tan(u)

sec^2(u)u'

Derivative of cot(u)

-csc^2(u)u'

Derivative of sec(u)

sec(u)tan(u)u'

Derivative of csc(u)

-csc(u)cot(u)u'

Derivative of arcsin(x)

1/sqrt(1-x^2)

Derivative of arccos(x)

-1/sqrt(1-x^2)

Derivative of arctan(x)

1/(1+x^2)

Derivative of arccot(x)

-1/(1+x^2)

Derivative of arcsec(x)

1/(|x|sqrt(x^2-1))

Derivative of arccsc(x)

-1/(|x|sqrt(x^2-1))

Derivative of arcsin(u)

u'/sqrt(1-u^2)

Derivative of arccos(u)

-u'/sqrt(1-u^2)

Derivative of arctan(u)

u'/(1+u^2)

Derivative of arccot(u)

-u'/(1+u^2)

Derivative of arcsec(u)

u'/(|u|sqrt(u^2-1))

Derivative of arccsc(u)

-u'/(|u|sqrt(u^2-1))

Derivative of e^x

e^x

Derivative of e^u

e^u u'

Derivative of a^x

a^x ln(a)

Derivative of a^u

a^u ln(a)u'

Derivative of ln(x)

1/x

Derivative of ln|x|

1/x

Derivative of ln|u|

u'/u

Derivative of log base a of x

1/(x ln(a))

Derivative of log base a of u

u'/(u ln(a))

Derivative of y^n when y depends on x

n y^(n-1) dy/dx

Derivative of sin(y) when y depends on x

cos(y) dy/dx

Derivative of cos(y) when y depends on x

-sin(y) dy/dx

Derivative of e^y when y depends on x

e^y dy/dx

Derivative of ln(y) when y depends on x

(1/y) dy/dx

Derivative of xy when y depends on x

x dy/dx + y

Logarithmic differentiation: y = f(x)^g(x)

y' = f(x)^g(x)[g'(x)ln(f(x)) + g(x)f'(x)/f(x)]

Derivative of x^x

x^x(ln(x)+1)

Derivative of x^(2x)

x^(2x)(2ln(x)+2)

Derivative of x^x e^x

x^x e^x(ln(x)+2)

Derivative of x^(sin(x))

x^(sin(x))[cos(x)ln(x) + sin(x)/x]

Derivative of x^(ln(x))

x^(ln(x))[(ln(x))^2 + ln(x)/x]

First derivative

f'(x); instantaneous rate of change or tangent slope

Second derivative

f''(x); derivative of the first derivative

Third derivative

f'''(x); derivative of the second derivative

Nth derivative

f^(n)(x); derivative taken n times

Derivative definition

f'(x) = lim as h->0 of [f(x+h)-f(x)]/h

Average rate of change on [a,b]

[f(b)-f(a)]/(b-a)

Circle area

A = pi r^2

Circle area rate of change

dA/dt = 2 pi r dr/dt

Circle circumference

C = 2 pi r

Sphere volume

V = (4/3)pi r^3

Sphere volume rate of change

dV/dt = 4 pi r^2 dr/dt

Cylinder volume

V = pi r^2 h

Cylinder volume rate with constant radius

dV/dt = pi r^2 dh/dt

Cone volume

V = (1/3)pi r^2 h

Pythagorean theorem

x^2 + y^2 = L^2

Related rates: first step

Write an equation relating the changing quantities.

Related rates: differentiation step

Differentiate both sides with respect to time t.

Related rates: why use the chain rule?

Variables such as x, y, r, and h depend on time.

Related rates: when to substitute values

Differentiate first, isolate the desired rate, then substitute known values.

Implicit differentiation: rule for y

When differentiating a function of y with respect to x, multiply by dy/dx.

Implicit differentiation: derivative of y^3

3y^2 dy/dx

Implicit differentiation: derivative of xy

x dy/dx + y

Implicit differentiation: main steps

Differentiate both sides, include dy/dx for y terms, collect dy/dx terms, and solve.

Logarithmic differentiation: main steps

Take ln of both sides, expand using log rules, differentiate implicitly, solve for y', and substitute y back.

Related rates: ladder equation

x^2 + y^2 = L^2, where L is the fixed ladder length

Related rates: cylinder height rate

dh/dt = (1/(pi r^2)) dV/dt

Power rule with negative exponents

d/dx[x^n] = n x^(n-1)

Power rule with fractional exponents

d/dx[x^n] = n x^(n-1)