Common Derivative Rules and Formulas
Expression	Derivative
y = xⁿ	dy/dx = n xⁿ⁻¹
y = a xⁿ	dy/dx = a n xⁿ⁻¹
f(x) = a xⁿ	f'(x) = a n xⁿ⁻¹
y = eˣ	dy/dx = eˣ
y = eᵃˣ	dy/dx = a eᵃˣ
y = aˣ	dy/dx = aˣ ln(a)
y = ln(x)	dy/dx = 1 / x
y = sin(Θ)	dy/dΘ = cos(Θ)
y = cos(Θ)	dy/dΘ = -sin(Θ)
y = tan(Θ)	dy/dΘ = sec²(Θ)
y = cot(Θ)	dy/dΘ = cosec²(Θ)
y = sec(Θ)	dy/dΘ = tan(Θ) sec(Θ) = sin(Θ) / cos²(Θ)
y = cosec(Θ)	dy/dΘ = -cot(Θ) cosec(Θ) = -cos(Θ) / sin²(Θ)
y = sin⁻¹(x / a)	dy/dx = 1 / √(a² - x²)
y = cos⁻¹(x / a)	dy/dx = -1 / √(a² - x²)
y = tan⁻¹(x / a)	dy/dx = a / (a² + x²)
y = cot⁻¹(x / a)	dy/dx = -a / (a² + x²)
y = sec⁻¹(x / a)	dy/dx = a / (x√(x² - a²))
y = cosec⁻¹(x / a)	dy/dx = -a / (x√(x² - a²))

Source: engineeringtoolbox.com

Key Derivative Rules Explained

This section breaks down the fundamental rules for finding derivatives, as summarized in the table above.

Power Rule

For a function $y = x^n$ , the derivative is $dy/dx = n x^{n-1}$ . This rule also applies when multiplied by a constant, such as $y = a x^n$ , giving $dy/dx = a n x^{n-1}$ .

Exponential & Logarithmic Rules

The derivative of the natural exponential function is itself: $d/dx (e^x) = e^x$ .
For a general exponential function $y = a^x$ , the derivative involves the natural logarithm: $dy/dx = a^x \ln(a)$ .
The derivative of the natural logarithm is: $d/dx (\ln x) = 1/x$ .

Trigonometric Function Derivatives

The derivatives of the basic trigonometric functions are:

$\frac{d}{d\Theta} \sin(\Theta) = \cos(\Theta)$
$\frac{d}{d\Theta} \cos(\Theta) = -\sin(\Theta)$
$\frac{d}{d\Theta} \tan(\Theta) = \sec^2(\Theta)$

Their reciprocals (cosecant, secant, cotangent) have derivatives that can be expressed in multiple forms, as shown in the table.

Inverse Trigonometric Function Derivatives

The derivatives of the inverse trigonometric functions yield algebraic expressions involving square roots and rational functions, which are essential for integration. For example:

$\frac{d}{dx} \sin^{-1}\left(\frac{x}{a}\right) = \frac{1}{\sqrt{a^2 - x^2}}$
$\frac{d}{dx} \tan^{-1}\left(\frac{x}{a}\right) = \frac{a}{a^2 + x^2}$

References

Original Source

Source: engineeringtoolbox.com