Common derivatives

Taking a derivative is also known as differentiating, because English.

Contents

Notation

The derivative function with respect to xx of f(x)f(x) is

 df dx= d dxf(x)=f′(x)\frac{\d f}{\d x} = \frac{\d }{\d x}f(x) = f^\prime(x)

The derivative evaluated at x=x0x = x_0 is

 df dx∣x=x0= d dxf(x)∣x=x0=f′(x0)\frac{\d f}{\d x}\Bigg\vert_{x=x_0} = \frac{\d}{\d x}f(x)\Bigg\vert_{x=x_0} = f^\prime(x_0)

Derivatives for common functions

This lists common derivatives for both SL and HL candidates. All derivatives are with respect to xx.

functionderivative function
constant, eg 55, k+1k+100
xn,n∈Q,n≠0x^n, n\in\mathbb{Q}, n\neq 0nxn−1nx^{n-1}
sin⁡x\sin xcos⁡x\cos x
cos⁡x\cos x−sin⁡x-\sin x
ex\e^xex\e^x
ln⁡x\ln x1x,x>0\frac 1x, x > 0
ln⁡∣x∣\ln \lvert x\rvert1x\frac 1x

Calculus of trig functions are always done in radians mode.

Properties

The rules apply to both the derivative functions and derivatives at a point, but only if the derivative exists. Where applicable, kk is a constant and does not depend on xx.

 d dx(f(x)+g(x))=f′(x)+g′(x)\frac{\d }{\d x} (f(x) + g(x)) = f^\prime(x) + g^\prime(x)
 d dxkf(x)=kf′(x)\frac{\d }{\d x} kf(x) = kf^\prime(x)
 d dx(f(x)+k)=f′(x)\frac{\d }{\d x} (f(x) + k) = f^\prime(x)

Chain rule

 d dxf(g(x))=f′(g(x))⋅g′(x)\frac{\d }{\d x} f(g(x)) = f^\prime(g(x)) \cdot g^\prime(x)

Example: Find

 d dx(−3x−12x2)(−3x+12x2)\frac{\d }{\d x}\left(-3x - \frac{1}{2x^2}\right)\left(-3x + \frac{1}{2x^2}\right)

This is a difference of squares. Notice everything is in the derivative because there is no ++ or −- outside of brackets or fractions. Let the function be f(x)f(x). Use difference of squares to simplify ff.

f(x)=9x2−14x4=9x2−14x−4f′(x)=9 d dxx2−14 d dxx−4=9(2x)−14(−4)x−5=18x+x−5■\begin{align*}f(x) &= 9x^2 - \frac{1}{4x^4} = 9x^2 - \frac 14x^{-4} \\ f^\prime(x) &= 9\frac{\d }{\d x}x^2 - \frac 14 \frac{\d }{\d x}x^{-4} \\ &= 9(2x) - \frac 14 (-4) x^{-5} \\ &= 18x + x^{-5} \qed \end{align*}

This would probably be only worth two marks so write only as much as you need to verify your work.

Useful formulas involving chain rule include

 d dxf(ax+b)=af′(x)\frac{\d }{\d x} f(ax + b) = af^\prime(x)
 d dx1f(x)=−f′(x)f(x)2\frac{\d }{\d x} \frac{1}{f(x)} = -\frac{f^\prime(x)}{f(x)^2}
 d dxef(x)=f′(x)ef(x)\frac{\d }{\d x} \e^{f(x)} = f'(x) \e^{f(x)}

Using Leibniz notation, the chain rule could be rewritten as

 dg dx= dg df⋅ df dx\frac{\d g}{\d x} = \frac{\d g}{\d f} \cdot \frac{\d f}{\d x}

This could be easier to remember as the “rates” can multiply with certain parts “cancelling out”.

Product rule

 d dxf(x)g(x)=g(x)f′(x)+f(x)g′(x)\frac{\d }{\d x} f(x)g(x) = g(x)f^\prime(x) + f(x)g^\prime(x)

Quotient rule

 d dxf(x)g(x)=g(x)f′(x)−f(x)g′(x)g(x)2\frac{\d }{\d x} \frac{f(x)}{g(x)} = \frac{g(x)f^\prime(x) - f(x)g^\prime(x)}{g(x)^2}

Tip: Remember that because g(x)g(x) is in the denominator, there is a negative sign from derivative of a negative exponent. So the term with g′(x)g^\prime(x) is associated with a minus.

Example: Find

 d dxtan⁡x\frac{\d }{\d x}\tan x

Use tan⁡x=sin⁡xcos⁡x\displaystyle\tan x = \frac{\sin x}{\cos x}.

 d dxtan⁡x= d dxsin⁡xcos⁡x=cos⁡x d dxsin⁡x−sin⁡x d dxcos⁡xcos⁡2x=cos⁡2x+sin⁡2xcos⁡2x=1cos⁡2x■\begin{align*}\frac{\d }{\d x} \tan x &= \frac{\d }{\d x} \frac{\sin x}{\cos x} \\ &= \frac{\cos x \frac{\d }{\d x} \sin x - \sin x \frac{\d }{\d x} \cos x}{\cos^2 x} \\ &= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \\ &= \frac{1}{\cos^2 x} \qed \end{align*}

Used the Pythagorean identity at the end.

Tips

  1. You are allowed both blue and black pens. Use one color for chain rule and the other color for product and quotient rules. For complicated derivatives, you may want to first copy the expression and circle parts of the expression needing each rule.
  2. Convert rational functions to polynomials raised to the power of negative exponents
  3. Give yourself copious space to work. You can always request additional answer booklets.
  4. Write faster and do less in your head.
  5. If there are multiple variables, which one are you differentiating with respect to? Does this variable appear in the base or in the exponent?
  6. For quotient rule, the derivative is zero if and only if the numerator is zero. Though when doing that, be sure to remove roots of the denominator.

HL tips

  1. Rewrite all powers and logs using base ee
  2. It may help to write all reciprocal trig functions in terms of sin⁡\sin and cos⁡\cos.