Complete Derivatives Table for Grade 12 — 35 Formulas to Know Before the Exam

What is a derivative? Why memorize the derivatives table?

Many grade 12 students fall into a very common trap when starting Calculus: they understand the theory of derivatives but haven't memorized the formulas. The result is that when they encounter problems finding maximum or minimum values of a function, or calculating integrals, they can't get the answer because they don't remember the derivative of sin x or e^x.

A derivative is the core tool of Calculus — it measures the rate of change of a function at a point. Geometrically, the derivative f'(x₀) is the slope of the tangent line to the graph y = f(x) at the point (x₀, f(x₀)). In the national high school math exam, derivatives appear directly or indirectly in roughly 20–25% of all questions: from curve sketching and finding extrema, to writing tangent line equations, indefinite and definite integrals.

Why is the derivatives table so important?

• Not knowing the derivatives table = not being able to solve extrema problems (which account for 4–6 questions in the exam).
• Integral problems require you to read the derivatives table in reverse to find antiderivatives.
• Chain rule derivatives appear extremely often in exams — if you don't remember the chain rule, you'll be wrong from the very first step.

This article compiles the complete derivatives table of 35 formulas — from basic derivatives to chain rule derivatives, trigonometric function derivatives, exponential and logarithmic function derivatives — with worked examples and quick memory tips.

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Basic Derivatives Table (15 Formulas)

Here are the 15 most fundamental derivative formulas that grade 12 students need to memorize. These are the foundation for differentiating any function.

Function f(x)	Derivative f'(x)	Condition
c (constant)	0	Derivative of a constant is always 0
x	1	—
x^n	n·x^(n−1)	n ∈ ℝ, n ≠ 0
√x	1 / (2√x)	x > 0
1/x	−1/x²	x ≠ 0
e^x	e^x	The only function that is its own derivative
a^x	a^x · ln(a)	a > 0, a ≠ 1
ln(x)	1/x	x > 0
log_a(x)	1 / (x·ln(a))	x > 0, a > 0, a ≠ 1
sin(x)	cos(x)	x in radians
cos(x)	−sin(x)	Note the negative sign!
tan(x)	1 / cos²(x)	x ≠ π/2 + kπ
cot(x)	−1 / sin²(x)	x ≠ kπ

Full formula notation:

$f(x)=x^n\Rightarrow f^{\prime }(x)=n{x}^{n-1}$

$f(x)=\sqrt{x}\Rightarrow f^{\prime }(x)=\frac{1}{2\sqrt{x}}$

Example 1: Find the derivative of f(x) = x⁵ − 3x² + 7.

$f^{\prime }(x)=5x^4-6x$

Example 2: Find the derivative of g(x) = √x + 1/x.

$g^{\prime }(x)=\frac{1}{2\sqrt{x}}-\frac{1}{x^2}$

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Chain Rule Derivatives Table (10 Formulas)

Chain rule derivatives are the type that appears most often in exams. The basic rule: if u = u(x) is a function of x, then:

$[f(u){]}^{\prime }=f^{\prime }(u)\cdot u^{\prime }$

That is: derivative of the outer function (keeping the inner function) multiplied by the derivative of the inner function.

Composite Function	Chain Rule Derivative	Condition
u^n	n·u^(n−1)·u'	Power chain rule
√u	u' / (2√u)	u > 0
1/u	−u'/u²	u ≠ 0
e^u	e^u · u'	Very common
a^u	a^u · ln(a) · u'	a > 0, a ≠ 1
ln(u)	u'/u	u > 0
log_a(u)	u' / (u·ln(a))	u > 0
sin(u)	cos(u)·u'	—
cos(u)	−sin(u)·u'	Note negative sign
tan(u)	u' / cos²(u)	—

Example 3: Find the derivative of f(x) = sin(3x² + 1).

Let u = 3x² + 1, so u' = 6x.

$f^{\prime }(x)=\cos (3x^2+1)\cdot 6x=6x\cos (3x^2+1)$

Example 4: Find the derivative of g(x) = e^(x² − 2x).

Let u = x² − 2x, so u' = 2x − 2.

$g^{\prime }(x)=e^{x^2-2x}\cdot (2x-2)$

Example 5: Find the derivative of h(x) = ln(x² + 1).

Let u = x² + 1, so u' = 2x.

$h^{\prime }(x)=\frac{2x}{x^2+1}$

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Trigonometric Function Derivatives in Detail

This section is particularly important because students often confuse the negative signs in the derivatives of cos x and cot x.

Basic Trigonometric Derivatives

Four formulas to memorize:

$(\sin x{)}^{\prime }=\cos x$

$(\cos x{)}^{\prime }=-\sin x$

$(\tan x{)}^{\prime }=\frac{1}{{\cos }^{2}x}$

$(\cot x{)}^{\prime }=-\frac{1}{{\sin }^{2}x}$

Remember: cos and cot have derivatives with negative signs — the two functions starting with "co" both have negative signs in their derivatives.

Composite Trigonometric Derivatives

Composite Form	Derivative
sin(f(x))	f'(x)·cos(f(x))
cos(f(x))	−f'(x)·sin(f(x))
tan(f(x))	f'(x) / cos²(f(x))
cot(f(x))	−f'(x) / sin²(f(x))

Example 6: Find the derivative of y = cos³(x).

Rewrite: y = (cos x)³. Let u = cos x, u' = −sin x.

$y^{\prime }=3(\cos x{)}^2\cdot (-\sin x)=-3\sin x{\cos }^{2}x$

Example 7: Find the derivative of y = tan(x²).

$y^{\prime }=\frac{(x^2{)}^{\prime }}{{\cos }^2(x^2)}=\frac{2x}{{\cos }^2(x^2)}$

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Exponential and Logarithmic Function Derivatives

This is the type of problem that many grade 12 students get wrong because they confuse e^x, a^x and their composite counterparts.

Exponential Functions

The most important formula to remember — the exponential function base e is its own derivative:

$(e^x{)}^{\prime }=e^x$

For exponential function with base a:

$(a^x{)}^{\prime }=a^x\ln a$

When a = e then ln(e) = 1, so (e^x)' = e^x · 1 = e^x — consistent.

Composite exponential derivative:

$(e^{u(x)}{)}^{\prime }=e^{u(x)}\cdot u^{\prime }(x)$

Example 8: Find the derivative of f(x) = 2^(3x+1).

$f^{\prime }(x)=2^{3x+1}\cdot \ln 2\cdot 3=3\ln 2\cdot 2^{3x+1}$

Logarithmic Functions

$(\ln x{)}^{\prime }=\frac{1}{x},({\log }_{a}x{)}^{\prime }=\frac{1}{x\ln a}$

Composite logarithmic derivatives:

$(\ln u{)}^{\prime }=\frac{u^{\prime }}{u},({\log }_{a}u{)}^{\prime }=\frac{u^{\prime }}{u\ln a}$

Example 9: Find the derivative of g(x) = ln((x+1)/(x−1)).

Use logarithm properties first: g(x) = ln(x+1) − ln(x−1).

$g^{\prime }(x)=\frac{1}{x+1}-\frac{1}{x-1}=\frac{-2}{x^2-1}$

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5 Quick Memory Tips for the Derivatives Table

Memorizing the derivatives table is not about rote learning — there are smart ways to remember longer and make fewer mistakes.

Tip 1: "Bring down the exponent, reduce by 1"

For x^n: bring the exponent down as a coefficient, then reduce the exponent by 1. (x⁵)' = 5x⁴. Simple and mechanical.

Tip 2: Sin-Cos cycle

Draw a cycle: sin → cos → −sin → −cos → sin → ...

• Derivative of sin is cos (one step forward).
• Derivative of cos is −sin (one step forward, with negative sign).
• Reverse when integrating.

Tip 3: Two functions with negative derivatives

Among the 4 trigonometric functions, cos and cot have derivatives with negative signs. Remember: "Co-negative" — the two functions starting with "co" (cosine, cotangent) both have negative signs in their derivatives.

Tip 4: e^x is "immortal"

e^x differentiated still gives e^x — it doesn't change. This is the unique property of the exponential function base e, extremely easy to remember.

Tip 5: Derivative of ln is "flip"

(ln x)' = 1/x — the derivative of the natural logarithm is simply "flipping" the inner function to the denominator. For composite: (ln u)' = u'/u — still flip u to the denominator, multiply by u' on top.

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Practice Problems (5 with Solutions)

Problem 1

Find the derivative of f(x) = x³ − 4x + 2/x.

Solution:

$f^{\prime }(x)=3x^2-4-\frac{2}{x^2}$

Problem 2

Find the derivative of f(x) = (2x² − 3x + 1)⁴.

Solution:

Let u = 2x² − 3x + 1, so u' = 4x − 3.

$f^{\prime }(x)=4u^3\cdot u^{\prime }=4(2x^2-3x+1{)}^3\cdot (4x-3)$

Problem 3

Find the derivative of f(x) = sin²(x) + cos²(x).

Solution:

We know sin²(x) + cos²(x) = 1 (Pythagorean identity), so f(x) = 1, therefore f'(x) = 0.

Can also compute directly:

$f^{\prime }(x)=2\sin x\cos x+2\cos x\cdot (-\sin x)=0$

Problem 4

Find the derivative of f(x) = e^(2x) · sin(x).

Solution (using product rule (uv)' = u'v + uv'):

Let u = e^(2x), v = sin(x). Then u' = 2e^(2x), v' = cos(x).

$f^{\prime }(x)=2e^{2x}\cdot \sin x+e^{2x}\cdot \cos x=e^{2x}(2\sin x+\cos x)$

Problem 5

Find the derivative of f(x) = ln(√(x² + 1)).

Solution:

Rewrite: f(x) = (1/2)·ln(x² + 1).

$f^{\prime }(x)=\frac{1}{2}\cdot \frac{2x}{x^2+1}=\frac{x}{x^2+1}$

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Frequently Asked Questions

Q1: What is the derivative of (sin x)²? Why isn't it 2·sin(x)?

(sin x)² is a composite function — the outer function is squaring, the inner function is sin(x). Applying the chain rule:

$[(\sin x{)}^2{]}^{\prime }=2\sin x\cdot (\sin x{)}^{\prime }=2\sin x\cdot \cos x=\sin 2x$

Many students mistakenly write 2·sin(x) because they forget to multiply by the derivative of the inner function (sin x)' = cos x.

Q2: When do I use the chain rule and when do I use the basic formula?

• Use the basic formula when the inner function is just x (e.g., sin(x), e^x, x^n).
• Use the chain rule when the inner function is a more complex expression than x (e.g., sin(3x+1), e^(x²), (x²+1)⁵).

Recognition tip: if you replace the inner function with the letter u and u is not simply x, then you need the chain rule.

Q3: What's the fastest way to memorize the derivatives table in one week?

The most effective method is practice with real problems, not reading and reciting. Every day, solve 10 basic derivative problems and 10 chain rule problems. After one week of active practice, the formulas will stick naturally. The Witza app has a set of derivative problems that lets you practice quickly and receive immediate feedback — perfect for daily review.

Q4: Why is the derivative of tan(x) equal to 1/cos²(x)?

Because tan(x) = sin(x)/cos(x), using the quotient rule (u/v)' = (u'v − uv')/v²:

$(\tan x{)}^{\prime }=\frac{\cos x\cdot \cos x-\sin x\cdot (-\sin x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}$

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Mastering derivatives correctly is the key to scoring well on the Calculus section of the national high school exam. If you want to practice the derivatives table systematically and test your knowledge through interactive practice exams, try Witza — the smart math learning app for high school students, with AI-powered problem solving and 1v1 battle mode with friends.