Chapter 2: Derivatives

The chain rule and logarithms

The chain rule is the rule for compositions, and once you have it almost every function in the course is in reach. The logarithm arrives with it: ln x is what you differentiate a composition through when the variable is in an exponent, and logarithmic differentiation turns an ugly product or power into a sum before any differentiating happens.

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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 $f(x) = (2x + 1)^{4}$ . Find $f'(0)$ .
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Introductory2 marks · 3 min
Let $F(x) = \sin\!\left(x^{2}\right)$ . Which of the following is $F'(x)$ ?
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Introductory1 mark · 2 min
Let $g(x) = \ln\!\left(x^{2}\right)$ for $x > 0$ . Find $g'(3)$ .
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Standard3 marks · 4 min
Let $y = \bigl(3x^{2} + 1\bigr)^{5}$ . Find $\dfrac{dy}{dx}$ at $x = 1$ .
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Standard3 marks · 4 min
Let $f(x) = \ln\bigl(x^{2} + 4\bigr)$ . Find $f'(2)$ .
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Standard3 marks · 4 min
Let $g(x) = x^{2}\,h\!\left(x^{3}\right)$ , where $h$ is a differentiable function with $h(8) = 3$ and $h'(8) = -1$ . Find $g'(2)$ .
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Standard3 marks · 4 min
Let $f(x) = \dfrac{e^{2x}}{1 + x^{2}}$ . Find $f'(0)$ .
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Challenging3 marks · 5 min
Let $g(x) = x^{2}\sin(3x)$ . Find $g'\!\left(\frac{\pi}{6}\right)$ .
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Challenging4 marks · 6 min
Let $f(x) = \bigl(x^{2} + 1\bigr)^{x}$ . Use logarithmic differentiation to find $f'(1)$ .
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Capstone5 marks · 8 min
The curve $y = x^{2}e^{-x}$ has exactly two points with a horizontal tangent. One of them is the origin. Find the $y$ -coordinate of the other.
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Differentiation rules
Power, sum, product and quotient rules, tangent lines and higher-order derivatives.
10 questions
Exponential and trigonometric functions
Derivatives of eˣ, aˣ, sine, cosine and tangent, and the limits behind them.
10 questions
Implicit differentiation and inverse trigonometric functions
Differentiating relations, tangent lines to curves, derivatives of inverse functions.
10 questions
Exponential growth and decay
The differential equation y′ = ky, half-lives, doubling times and Newton’s law of cooling.
10 questions
Linear approximation and Taylor polynomials
Tangent-line estimates, quadratic approximations, Taylor polynomials and error bounds.
10 questions
Optimization
Building an objective function and justifying the extreme value.
10 questions
Curve sketching
Increase, concavity, inflection points, asymptotes and full sketches.
10 questions
L’Hôpital’s rule
Indeterminate forms 0/0, ∞/∞, 0·∞, ∞−∞ and 1^∞, and when the rule does not apply.
10 questions
Antiderivatives
Reversing differentiation, the constant of integration and initial conditions.
10 questions

Practice questions

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