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Longhand

Chapter 2: Derivatives

Exponential and trigonometric functions

Two families of functions are their own derivatives up to a factor: exponentials, and sine and cosine taken together. Everything about them follows from two limits, the one that defines e and the one that says sin x behaves like x near zero. Most exam questions are the product and quotient rules applied to these functions, or a tangent line at a value where they are easy to evaluate.

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Practice questions

Every question below is original material written for Longhand. Open a solution to read the full working, or answer it yourself to have it counted towards your progress.

  • Introductory1 mark · 2 min

    Let f(x)=4ex−2sin⁡xf(x) = 4e^{x} - 2\sin x. Find f′(0)f'(0).

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  • Introductory1 mark · 2 min

    Which of the following is the derivative of g(x)=3xg(x) = 3^{x}?

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  • Introductory2 marks · 3 min

    Which of the following is the derivative of h(x)=x2exh(x) = x^{2}e^{x}?

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  • Standard3 marks · 4 min

    Find the yy-intercept of the line tangent to y=sin⁡x+3xy = \sin x + 3x at x=πx = \pi.

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  • Standard3 marks · 5 min

    Find every xx in [0,2π][0, 2\pi] at which the graph of f(x)=sin⁡x+cos⁡xf(x) = \sin x + \cos x has a horizontal tangent, and enter the sum of those values.

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  • Standard3 marks · 4 min

    Let f(x)=ex1+cos⁡xf(x) = \dfrac{e^{x}}{1 + \cos x}. Find f′(0)f'(0).

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  • Standard2 marks · 3 min

    On which interval is f(x)=xexf(x) = xe^{x} increasing?

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  • Challenging4 marks · 6 min

    Find the constants aa and bb for which the function below is differentiable everywhere, and enter a+ba + b.

    f(x)={ax+b,x<03sin⁡x+4cos⁡x,x≥0f(x) = \begin{cases} ax + b, & x < 0 \\[4pt] 3\sin x + 4\cos x, & x \geq 0 \end{cases}
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  • Challenging3 marks · 5 min

    Evaluate the limit by recognising it as a derivative.

    lim⁡x→π/6sin⁡x−12x−π6\lim_{x \to \pi/6} \frac{\sin x - \tfrac{1}{2}}{x - \tfrac{\pi}{6}}
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  • Capstone5 marks · 9 min

    The function below is differentiable everywhere. Find aa.

    f(x)={ax+b,x<05cos⁡x3+sin⁡x+cos⁡x,x≥0f(x) = \begin{cases} ax + b, & x < 0 \\[6pt] \dfrac{5\cos x}{3 + \sin x + \cos x}, & x \geq 0 \end{cases}
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Related topics

  • Differentiation rules

    Power, sum, product and quotient rules, tangent lines and higher-order derivatives.

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  • The chain rule and logarithms

    Compositions, nested rules, the natural logarithm and logarithmic differentiation.

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  • Exponential growth and decay

    The differential equation y′ = ky, half-lives, doubling times and Newton’s law of cooling.

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  • Antiderivatives

    Reversing differentiation, the constant of integration and initial conditions.

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