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Formulas: calculus, grade 11 (5 units)

Rules of differentiation and the tangent line

  • The derivative of a power
    (xn)′\displaystyle (x^n)' =\displaystyle = nxn−1\displaystyle nx^{n-1}
  • The derivative of a product
    [f(x)⋅g(x)]′\displaystyle [f(x)\cdot g(x)]' =\displaystyle = f′(x)⋅g(x)+f(x)⋅g′(x)\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)
  • The derivative of a quotient
    [f(x)g(x)]′\displaystyle \left[\frac{f(x)}{g(x)}\right]' =\displaystyle = f′(x)⋅g(x)−f(x)⋅g′(x)[g(x)]2\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}
  • The derivative of a composite function – the outer derivative times the inner derivative
    [f(u(x))]′\displaystyle [f(u(x))]' =\displaystyle = f′(u)⋅u′(x)\displaystyle f'(u)\cdot u'(x)
  • The derivative of 1x\displaystyle \frac{1}{x}
    (1x)′\displaystyle \left(\frac{1}{x}\right)' =\displaystyle = −1x2\displaystyle -\frac{1}{x^2}
  • The tangent line at the point (x1,y1)\displaystyle (x_1,y_1) on the graph
    y−y1=f′(x1)(x−x1)\displaystyle y-y_1=f'(x_1)(x-x_1)

The second derivative

  • The second derivative – the derivative of the derivative
    f′′(x)\displaystyle f''(x) =\displaystyle = (f′(x))′\displaystyle \left(f'(x)\right)'
  • The function is concave up (∪\displaystyle \cup)
    f′′(x)>0\displaystyle f''(x)>0
  • The function is concave down (∩\displaystyle \cap)
    f′′(x)<0\displaystyle f''(x)<0
  • A possible inflection point – check that the sign of f′′\displaystyle f'' changes there
    f′′(x0)=0\displaystyle f''(x_0)=0
  • A minimum point
    f′(x0)=0, f′′(x0)>0\displaystyle f'(x_0)=0,\ f''(x_0)>0
  • A maximum point
    f′(x0)=0, f′′(x0)<0\displaystyle f'(x_0)=0,\ f''(x_0)<0
  • Acceleration – the derivative of the speed
    a(t)=v′(t)=s′′(t)\displaystyle a(t)=v'(t)=s''(t)

Asymptotes

  • A vertical asymptote – f(x)\displaystyle f(x) grows or falls without end as x\displaystyle x approaches a\displaystyle a
    x=a\displaystyle x=a
  • A horizontal asymptote – f(x)\displaystyle f(x) approaches b\displaystyle b as x→∞\displaystyle x\to\infty or as x→−∞\displaystyle x\to-\infty
    y=b\displaystyle y=b

Roots

  • The root of a square
    x2\displaystyle \sqrt{x^2} =\displaystyle = ∣x∣\displaystyle |x|
  • The derivative of x\displaystyle \sqrt{x}
    (x)′\displaystyle (\sqrt{x})' =\displaystyle = 12x\displaystyle \frac{1}{2\sqrt{x}}x>0\displaystyle x>0
  • The derivative of the root of a function
    (f(x))′\displaystyle (\sqrt{f(x)})' =\displaystyle = f′(x)2f(x)\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}f(x)>0\displaystyle f(x)>0

Radians and periods

  • Half a turn in radians
    π\displaystyle \pi =\displaystyle = 180∘\displaystyle 180^{\circ}
  • The length of an arc of α\displaystyle \alpha radians in a circle of radius R\displaystyle R
    l\displaystyle l =\displaystyle = αR\displaystyle \alpha R
  • The period of sin⁡x\displaystyle \sin x and of cos⁡x\displaystyle \cos x is 2π\displaystyle 2\pi
    sin⁡(x+2π)\displaystyle \sin(x+2\pi) =\displaystyle = sin⁡x\displaystyle \sin x
  • The period of tan⁡x\displaystyle \tan x is π\displaystyle \pi
    tan⁡(x+π)\displaystyle \tan(x+\pi) =\displaystyle = tan⁡x\displaystyle \tan x
  • The limit behind the derivative of sin⁡x\displaystyle \sin x (x\displaystyle x in radians)
    lim⁡x→0sin⁡xx=1\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1

Trigonometric identities

  • The Pythagorean identity
    sin⁡2α+cos⁡2α\displaystyle \sin^2\alpha+\cos^2\alpha =\displaystyle = 1\displaystyle 1
  • Tangent
    tan⁡α\displaystyle \tan\alpha =\displaystyle = sin⁡αcos⁡α\displaystyle \frac{\sin\alpha}{\cos\alpha}
  • Sine – an odd function
    sin⁡(−α)\displaystyle \sin(-\alpha) =\displaystyle = −sin⁡α\displaystyle -\sin\alpha
  • Cosine – an even function
    cos⁡(−α)\displaystyle \cos(-\alpha) =\displaystyle = cos⁡α\displaystyle \cos\alpha
  • The sine of the supplement to π\displaystyle \pi
    sin⁡(π−α)\displaystyle \sin(\pi-\alpha) =\displaystyle = sin⁡α\displaystyle \sin\alpha
  • The cosine of the complement to π2\displaystyle \frac{\pi}{2}
    cos⁡(π2−α)\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right) =\displaystyle = sin⁡α\displaystyle \sin\alpha
  • The sine of a double angle
    sin⁡2α\displaystyle \sin2\alpha =\displaystyle = 2sin⁡α⋅cos⁡α\displaystyle 2\sin\alpha\cdot\cos\alpha
  • The cosine of a double angle
    cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\displaystyle \cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha
  • The difference of two sines
    sin⁡α−sin⁡β\displaystyle \sin\alpha-\sin\beta =\displaystyle = 2cos⁡α+β2⋅sin⁡α−β2\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}
  • The sine of a sum and of a difference
    sin⁡(α±β)\displaystyle \sin(\alpha\pm\beta) =\displaystyle = sin⁡α⋅cos⁡β±cos⁡α⋅sin⁡β\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta

Derivatives of the trigonometric functions

  • The derivative of sine (x\displaystyle x in radians)
    (sin⁡x)′\displaystyle (\sin x)' =\displaystyle = cos⁡x\displaystyle \cos x
  • The derivative of cosine
    (cos⁡x)′\displaystyle (\cos x)' =\displaystyle = −sin⁡x\displaystyle -\sin x
  • The derivative of tangent
    (tan⁡x)′\displaystyle (\tan x)' =\displaystyle = 1cos⁡2x\displaystyle \frac{1}{\cos^2x}