Calculus I Derivative Functions + Basic & Hyperbolic Trig + MVT/RT/IVT

For Bethel Gateway Exam + Exam #2 + Final

Derivative of tan⁻¹(x) or (arctan)

1/1+x²

Derivative of sin⁻¹(x) or (arcsin)

1/√1-x²

Derivative of (ƒ⁻¹)(x)

1/ƒ’(ƒ⁻¹(x))

Derivative of e∧ln(x)

x

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec²(x) or (sec(x))²

Derivative of cot(x)

(-csc(x))²

Derivative of sec(x)

(sec(x))(tan(x))

Derivative of csc(x)

(-csc(x))(cot(x))

Derivative of ln(x)

1/x

What is tan(x)?

sin(x)/cos(x)

What is cot(x)?

cos(x)/sin(x)

What is sec(x)?

1/cos(x)

What is csc(x)?

1/sin(x)

Derivative of ƒ(g(x))

ƒ’(g(x))×g’(x)

Derivative of ƒ/g

(ƒ’g-ƒg’)/g²

Derivative of a^x

(lna)a^x

Derivative of e^x

e^x

Derivative of xⁿ

nxⁿ⁻¹

Derivative of cƒ(x)

cƒ’(x)

Derivative of [ƒ(x) ± g(x)]

ƒ’(x) ± g’(x)

Rolle’s Theorem

If ƒ is continuous on [a,b], differentiable on (a,b), and ƒ(a)=ƒ(b), then there exists a value c somewhere between a and b such that ƒ’(c)=0.

Mean Value Theorem

If ƒ is continuous on [a,b] and differentiable on (a,b). then there exists a value between a and b such that ƒ’(c)=(ƒ(b)-ƒ(a))/(b-a).

Derivative of tanh(x)

1/(cosh(x))²

Derivative of sinh(x)

cosh(x)

Derivative of cosh(x)

sinh(x)

What is tanh(x)

((e^x)-(e^-x))/((e^x)+(e^-x)) or sinh(x)/cosh(x)

What is cosh(x)

((e^x)+(e^-x))/2

What is sinh(x)

(e^x)-(e^-x))/2

Error in the approximation of L(x)

ƒ(x)-L(x) or E(x)=ƒ(x)-[ƒ(a)+ƒ’(a)(x-a)]

Intermediate Value Theorem

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k

Limit definition of a derivative

f’(x) = lim as h→0 (f(x+h) - f(x))/h