Differentiation (FP3)

Q: Do I need to memorise standard derivatives for this topic?

No, all standard derivatives are provided in the Edexcel IAL Further Maths formula book, but recalling them quickly will save you exam time.

Q: Can I leave my differentiation results unsimplified?

Full marks are only awarded for fully simplified results, so you must combine terms, rationalise surds and cancel common factors where possible.

Edexcel International A-Level Further Mathematics· 2018 Spec Issue 3 FP3 3.1-3.2· 25 min read

1. Differentiating Hyperbolic Functions★★☆☆☆⏱ 8 min

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You will use the standard derivatives of hyperbolic functions (given in your formula book) combined with core pure maths differentiation rules to evaluate derivatives of expressions involving $sinh x$ , $cosh x$ and $tanh x$ .

\frac{d}{dx}\left(\sinh x\right) = \cosh x, \quad \frac{d}{dx}\left(\cosh x\right) = \sinh x, \quad \frac{d}{dx}\left(\tanh x\right) = \text{sech}^2 x

📐 Worked Example

Differentiate $f (x) = x sinh^{2} 3 x$

1
Recognise the function is a product of $u = x$ and $v = sinh^{2} 3 x$ , so apply the product rule: $f^{'} (x) = u^{'} v + u v^{'}$
2
Calculate $u^{'}$ : $u^{'} = \frac{d}{d x} (x) = 1$
3
Calculate $v^{'}$ using the chain rule twice: let $w = 3 x$ , so $v = (sinh w)^{2}$ , so $v^{'} = 2 sinh w \cdot cosh w \cdot 3 = 6 sinh 3 x cosh 3 x$
4
Substitute into the product rule: $f^{'} (x) = sinh^{2} 3 x + 6 x sinh 3 x cosh 3 x$
5
Factor the final result: $f^{'} (x) = sinh 3 x (sinh 3 x + 6 x cosh 3 x)$

Exam tip:

Always factor out common terms from your final result to meet simplification requirements for full marks.

2. Differentiating Inverse Trigonometric Functions★★★☆☆⏱ 8 min

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Standard derivatives of inverse trigonometric functions are provided in your formula book. Combine these with core differentiation rules to evaluate derivatives of more complex inverse trig expressions.

\frac{d}{dx}\left(\arcsin x\right) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\left(\arccos x\right) = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\left(\arctan x\right) = \frac{1}{1+x^2}

📐 Worked Example

Differentiate $g (x) = arcsin x + x 1 - x^{2}$

1
Split the function into two terms to differentiate individually: $g^{'} (x) = \frac{d}{d x} (arcsin x) + \frac{d}{d x} (x 1 - x^{2})$
2
Differentiate the first term using the standard result: $\frac{d}{d x} (arcsin x) = \frac{1}{1 - x ^{2}}$
3
Apply the product rule to the second term: let $u = x$ , $v = (1 - x^{2})^{1/2}$ , so $u^{'} = 1$ , $v^{'} = - \frac{x}{1 - x ^{2}}$
4
Evaluate the second term derivative: $1 - x^{2} - \frac{x ^{2}}{1 - x ^{2}}$
5
Combine and simplify all terms: $g^{'} (x) = \frac{1 + ( 1 - x ^{2} ) - x ^{2}}{1 - x ^{2}} = 2 1 - x^{2}$

Exam tip:

Combine terms with square root denominators into a single fraction before simplifying to avoid arithmetic errors.

3. Differentiating Inverse Hyperbolic Functions★★★☆☆⏱ 7 min

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Standard derivatives of inverse hyperbolic functions are also provided in your formula book. Apply the same chain, product and quotient rules you use for other function types to these expressions.

\frac{d}{dx}\left(\text{arsinh }x\right) = \frac{1}{\sqrt{1+x^2}}, \quad \frac{d}{dx}\left(\text{arcosh }x\right) = \frac{1}{\sqrt{x^2-1}}, \quad \frac{d}{dx}\left(\text{artanh }x\right) = \frac{1}{1-x^2}

📐 Worked Example

Differentiate $h (x) = artanh (x^{2})$

1
Recognise this is a composite function, so apply the chain rule: let $u = x^{2}$ , so $h (x) = artanh u$
2
Apply the chain rule formula: $h^{'} (x) = \frac{d}{d u} (artanh u) \cdot \frac{d u}{d x}$
3
Substitute the standard derivative: $\frac{d}{d u} (artanh u) = \frac{1}{1 - u ^{2}}$
4
Calculate $\frac{d u}{d x} = 2 x$
5
Substitute back and simplify: $h^{'} (x) = \frac{2 x}{1 - x ^{4}}$

Exam tip:

Always confirm the domain of the inverse function when evaluating derivatives, e.g. $arcosh x$ is only defined for $x \geq 1$ .

4. Common Pitfalls

Wrong move:

Forgetting the negative sign in the derivative of $arccos x$

Why:

Mixing up the signs of $arcsin x$ and $arccos x$ derivatives

Correct move:

Double-check the sign for $arccos x$ and cross-reference the formula book if unsure

Wrong move:

Differentiating $sinh^{2} x$ as $2 sinh x$ instead of $2 sinh x cosh x$

Why:

Missing the inner derivative of the hyperbolic function when applying the chain rule

Correct move:

Apply the chain rule twice for powers of hyperbolic functions: once for the power, once for the hyperbolic term

Wrong move:

Leaving results with square root denominators unsimplified

Why:

Exam mark schemes require fully simplified results for full marks

Correct move:

Combine fractions with surd denominators, cancel common factors and rationalise before writing your final answer

Wrong move:

Writing $1/ 1 - x^{2}$ as the derivative of $arsinh x$ instead of $1/ 1 + x^{2}$

Why:

Mixing up inverse trig and inverse hyperbolic derivative forms

Correct move:

Remember inverse trig derivatives have $1 - x^{2}$ under the root, while $arsinh x$ has $1 + x^{2}$

Wrong move:

Differentiating $tanh 3 x$ as $sech^{2} 3 x$ instead of $3 sech^{2} 3 x$

Why:

Rushing through chain rule steps for linear inner functions

Correct move:

Write the inner function explicitly when applying the chain rule to avoid missing its derivative

5. Quick Reference Cheatsheet

Function	Standard Derivative	Key Tip
$sinh x$	$cosh x$	No sign change
$cosh x$	$sinh x$	No sign change
$tanh x$	$sech^{2} x$	Squared hyperbolic secant
$arcsin x$	$1/ 1 - x^{2}$	Positive denominator
$arccos x$	$- 1/ 1 - x^{2}$	Negative numerator
$arctan x$	$1/ (1 + x^{2})$	No square root
$arsinh x$	$1/ 1 + x^{2}$	Plus sign under root
$arcosh x$	$1/ x^{2} - 1$	$x^{2} - 1$ under root
$artanh x$	$1/ (1 - x^{2})$	No square root, $∣ x ∣ < 1$

6. Frequently Asked

Do I need to memorise standard derivatives for this topic?

No, all standard derivatives are provided in the Edexcel IAL Further Maths formula book, but recalling them quickly will save you exam time.

Can I leave my differentiation results unsimplified?

Full marks are only awarded for fully simplified results, so you must combine terms, rationalise surds and cancel common factors where possible.

Going deeper

formula_bookEdexcel IAL Further Maths Formula BookRefer to FP3 Differentiation section for standard derivatives

What's Next

Now you have mastered FP3 differentiation of hyperbolic and inverse functions, you are ready to move on to FP3 Integration, which uses the reverse of the derivative rules you have learned here. This differentiation knowledge will also form the foundation for solving differential equations in FP3, as well as calculus applications in other Further Maths units like FP2 and Mechanics 3. Make sure you practice plenty of past paper questions on this topic to build speed and accuracy, as differentiation questions are often combined with other topics in longer exam questions worth 5-8 marks. Regular practice will help you avoid the common pitfalls listed earlier and ensure you can simplify results quickly under exam conditions.