Derivative Formulas: Full List with Proofs (Class 12 & JEE)

Function	Derivative
d/dx(sin x)	cos x
d/dx(cos x)	−sin x
d/dx(tan x)	sec²x
d/dx(cot x)	−cosec²x
d/dx(sec x)	sec x tan x
d/dx(cosec x)	−cosec x cot x
d/dx(eˣ)	eˣ
d/dx(aˣ)	aˣ log a
d/dx(log x)	1/x
d/dx(xⁿ)	n xⁿ⁻¹
d/dx(√x)	1/(2√x)
d/dx(1/x)	−1/x²
d/dx(sin⁻¹x)	1/√(1 − x²)
d/dx(cos⁻¹x)	−1/√(1 − x²)
d/dx(tan⁻¹x)	1/(1 + x²)
d/dx(cot⁻¹x)	−1/(1 + x²)
d/dx(sec⁻¹x)	1/(\|x\|√(x² − 1))
d/dx(xˣ)	xˣ(1 + log x)
d/dx(logₐx)	1/(x log a)
d/dx(sin²x)	sin 2x
d/dx(log(sin x))	cot x
d/dx(\|x\|)	x/\|x\| (x ≠ 0)

Function

Derivative

cos x

−sin x

sec²x

−cosec²x

sec x tan x

−cosec x cot x

eˣ

aˣ log a

1/x

n xⁿ⁻¹

1/(2√x)

−1/x²

1/√(1 − x²)

−1/√(1 − x²)

1/(1 + x²)

−1/(1 + x²)

1/(|x|√(x² − 1))

xˣ(1 + log x)

1/(x log a)

sin 2x

cot x

x/|x| (x ≠ 0)

The rules you combine these with

Product rule: (uv)′ = u′v + uv′
Quotient rule: (u/v)′ = (u′v − uv′)/v²
Chain rule: d/dx f(g(x)) = f′(g(x)) × g′(x)
Logarithmic differentiation for xˣ or products of many terms: take log first, then differentiate.

Questions students ask

What is the derivative of tan x?

sec²x.

What is the derivative of sin⁻¹x?

1/√(1 − x²), for −1 < x < 1.

What is the derivative of xˣ?

xˣ(1 + log x), using logarithmic differentiation.

Derivative formulas

The rules you combine these with

Questions students ask

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