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AI HL · AHL 2

Further Derivatives

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AHL 2.1

Differentiating powers, trig, e^x and ln x

AHL 2.1

Differentiating powers, trig, e^x and ln x

Target: I can differentiate \(x^n\) and \(kx^n\) when n is a positive or negative fraction. (+4 more)

By the end of this section

Key points

Powers of x

  • For any rational n: if \(f(x)=x^n\), then \(f'(x)=nx^{n-1}\). The rule is the same for fractions and negative numbers.
  • Rewrite before you differentiate: \(\sqrt[3]{x}=x^{1/3}\), \(\dfrac{1}{\sqrt{x}}=x^{-1/2}\), \(\dfrac{10}{x^n}=10x^{-n}\).
  • Split a fraction with a sum on top into separate powers, using \(\dfrac{x^a}{x^b}=x^{a-b}\). Then differentiate term by term.
  • Subtract the powers when you divide. \(x^3\div x^{1/2}=x^{5/2}\), not \(x^{7/2}\).

Trigonometric functions

  • \(\dfrac{d}{dx}\sin x=\cos x\), \(\dfrac{d}{dx}\cos x=-\sin x\), \(\dfrac{d}{dx}\tan x=\dfrac{1}{\cos^2 x}\).
  • These are true only when x is in radians. Set the calculator to radians first.
  • Differentiate term by term. A constant on its own has derivative 0.

e^x and ln x

  • \(\dfrac{d}{dx}e^x=e^x\). The function is its own derivative.
  • \(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\), for x > 0.
  • For the logarithm of a power, use \(\ln x^m=m\ln x\) first. Then \(\dfrac{d}{dx}\ln x^m=\dfrac{m}{x}\).

On the calculator

Numerical derivative at a point

Menu > Calculus > Numerical Derivative at a Point

  • Use this to check the value of a derivative at a given x.
  • Use the menu. Typing the shortcut symbol for d/dx can give the error "Variable is not defined".
  • Compare the number with your own derivative, substituted at the same x.

Screenshot and video to come

Radian mode

  • The derivatives of sin x, cos x and tan x are true only in radians.
  • Check the angle setting before you evaluate a trigonometric derivative.
  • A wrong answer such as 2.86 instead of −4.89 often means the calculator was in degrees.

Screenshot and video to come

Gradient equal to zero

  • Write \(f'(x)=0\) and solve it with the calculator when it cannot be solved by hand.
  • Give the answer to 3 significant figures.
  • Reject a solution outside the domain, such as x ≤ 0 for \(\ln x\).

Screenshot and video to come

Explore: from f to f′

Drag the orange point along the graph of f. Each place you stop leaves a dot on the lower graph at the height of the gradient. Trace f′ yourself, then press Show f′ to check.


x
f(x)
Dot on the lower graph

For sin x, x is in radians.

Drag the point along f.

  1. Before you drag, predict the shape of f′ for x². Where will the dots cross the x-axis? Why there?
  2. On the cubic, f has two turning points. Where are they, and what is true of f′ at those values of x?
  3. Where is f steepest? Where is the dot highest or lowest on the lower graph?
  4. On the quartic, how many times will f′ cross the x-axis? Check with Show f′.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A root as a power

Given \(f(x)=\sqrt[5]{x}\), find \(f'(32)\).

Working
  1. Write as a power

    \(f(x)=x^{1/5}\)

    A root is a fractional power. Rewrite before you differentiate.

  2. Power rule

    \(f'(x)=\frac{1}{5}x^{1/5-1}=\frac{1}{5}x^{-4/5}\)

    Bring the power down and subtract 1 from it.

  3. Substitute

    \(f'(32)=\frac{1}{5}\times 32^{-4/5}=\frac{1}{5}\times\frac{1}{16}\) = 0.0125

    32 = 2⁵, so \(32^{4/5}=2^4=16\).

  4. Check

    From the calculator: numerical derivative of \(x^{1/5}\) at x = 32: 0.0125

    Use the menu, not the shortcut symbols.

Video walkthrough coming
Split a fraction

Given \(y=\dfrac{4x^3+6}{\sqrt{x}}\), find the gradient of the curve at x = 4.

Working
  1. Split

    \(y=\frac{4x^3}{x^{1/2}}+\frac{6}{x^{1/2}}\)

    Divide each term in the numerator by the denominator.

  2. Powers

    \(y=4x^{5/2}+6x^{-1/2}\)

    Subtract powers when you divide: 3 − 1/2 = 5/2.

  3. Differentiate

    \(\frac{dy}{dx}=10x^{3/2}-3x^{-3/2}\)

    4 × 5/2 = 10 and 6 × (−1/2) = −3.

  4. Substitute

    \(\frac{dy}{dx}=10\times 8-3\times\frac{1}{8}\) = 79.625

    \(4^{3/2}=8\).

Video walkthrough coming
Sine and cosine

Given \(f(x)=3\sin x+4\cos x\), find \(f'(2)\), with x in radians.

Working
  1. Differentiate

    \(f'(x)=3\cos x-4\sin x\)

    The derivative of sin x is cos x. The derivative of cos x is −sin x.

  2. Radian mode

    Calculator set to radians.

    The rules for the derivatives are true only for radians.

  3. Substitute

    \(f'(2)=3\cos 2-4\sin 2\) = −4.89

    From the calculator.

Video walkthrough coming
The tan function

Given \(y=3\tan x-x\), find \(\dfrac{dy}{dx}\) when x = 1 (radians).

Working
  1. Differentiate

    \(\frac{dy}{dx}=\frac{3}{\cos^2 x}-1\)

    The derivative of tan x is \(\frac{1}{\cos^2 x}\). The derivative of x is 1.

  2. Substitute

    \(\frac{dy}{dx}=\frac{3}{\cos^2 1}-1\) = 9.28

    From the calculator, in radian mode.

Video walkthrough coming
Log of a power

Given \(y=2\ln x^3\), find \(y'\) when x = 0.5.

Working
  1. Log law

    \(y=2\ln x^3=6\ln x\)

    \(\ln x^m=m\ln x\): the power comes to the front.

  2. Differentiate

    \(y'=\frac{6}{x}\)

    The derivative of ln x is 1/x.

  3. Substitute

    \(y'=\frac{6}{0.5}\) = 12

    Dividing by 0.5 doubles.

Video walkthrough coming
Exponential and log

The curve \(y=4e^x-3\ln x\) is defined for x > 0. Find its gradient at x = 1.

Working
  1. Differentiate

    \(\frac{dy}{dx}=4e^x-\frac{3}{x}\)

    The derivative of \(e^x\) is \(e^x\). The derivative of ln x is 1/x.

  2. Substitute

    \(\frac{dy}{dx}=4e^1-\frac{3}{1}\) = 7.87

    From the calculator: \(4e-3\).

Video walkthrough coming