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IB Mathematics: Applications and Interpretation HL · AHL 2: Further Derivatives · AHL 2.1

Differentiating powers, trig, e^x and ln x Problem set A Answers

Skills

    1. (a) \(f'(x)=\frac{5}{3}x^{2/3}\)
    2. (b) \(f'(x)=-9x^{-7/4}\)
    1. (a) \(f'(x)=2x^{-2/3}\)
    2. (b) \(f'(x)=\frac{9}{4}x^{-5/4}\)
    1. (a) \(\frac{dy}{dx}=8x^{1/3} - \frac{10}{3}x^{-5/3}\)
    2. (b) \(\frac{dy}{dx}=-\frac{7}{5}x^{-6/5} - \frac{8}{5}x^{-1/5}\)
    1. (a) \(f'(x)=5\cos x + 3\sin x\)
    2. (b) \(f'(x)=\frac{1}{3}\sin x + 6\cos x\)
    1. (a) \(f'(x)=\frac{4}{\cos^2 x}\)
    2. (b) \(f'(x)=2x + \frac{1}{\cos^2 x}\)
    1. (a) \(y'=\frac{6}{x}\)
    2. (b) \(y'=-\frac{5}{x} + 3e^x\)
    1. (a) \(y'=\frac{14}{x}\)
    2. (b) \(y'=\frac{6}{x}\)

In context

    1. (a) \(\frac{dh}{dt}=15t^{-1/4}\)
    2. (b) 7.5 cm per day
    1. (a) \(\frac{dd}{dt}=2\cos t - \frac{3}{2}\sin t\)
    2. (b) −2.19 m per hour
    1. (a) \(\frac{dy}{d\theta}=\frac{25}{\cos^2 \theta}\)
    2. (b) 64.7 m per radian
    1. (a) \(V=12 + 15\ln t;\ \frac{dV}{dt}=\frac{15}{t}\)
    2. (b) 3.75 thousand dollars per year

Stretch

    1. (a) \(f'(x)=\frac{3}{2}x^{1/2} - 3x^{-1/2}\)
    2. (b) x = 2
    1. (a) \(\frac{dy}{dx}=e^x - \frac{4}{x}\)
    2. (b) x = 1.20

jmaths.xyz · AI HL · AHL 2.1 Differentiating powers, trig, e^x and ln x