Basic Derivatives
Knowing the basic derivatives is the essential first step toward solving more advanced calculus problems.
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( c \) | \( 0 \) |
| \( x \) | \( 1 \) |
| \( x^n \) | \( n \cdot x^{n-1} \) |
| \( \frac{1}{x} = x^{-1} \) | \( - \frac{1}{x^2} \) |
| \( \sqrt[n]{x} = x^{ \frac{1}{n} } \) | \( \frac{1}{x \sqrt[n]{x^{n-1}}} \) |
| \( e^x \) | \( e^x \) |
| \( \ln(x) \) | \( \frac{1}{x} \) |
| \( \log_b(x) \) | \( \frac{1}{ x} \log_b e \) |
| \( \sin(x) \) | \( \cos(x) \) |
| \( \cos(x) \) | \( -\sin(x) \) |
| \( \tan(x) \) | \( \frac{1}{ \cos^2(x) } = 1+ \tan^2 x \) |
| \( \cot(x) \) | \( - \frac{1}{ \sin^2(x) } = -1 -\cot^2 x \) |
| \( \arcsin(x) \) | \( \frac{1}{ \sqrt{1-x^2} } \) |
| \( \sec(x) \) | \( \sec(x) \cdot \tan(x) \) |
| \( \arccos(x) \) | \( - \frac{1}{ \sqrt{1-x^2} } \) |
| \( \arctan(x) \) | \( \frac{1}{ 1+x^2 } \) |
| \( a^x \) | \( a^x \ln a \ \ \ (a>0) \) |
| \( x^x \) | \( x^x (1+ \log x) \) |
| \( e^{f(x)} \) | \( e^{f(x) } \cdot f'(x) \) |
| \( f(x)^{\alpha} \) | \( \alpha f(x)^{ \alpha -1 } f'(x) \) |
| \( f(g(x)) \) | \( f'(g(x)) \cdot g'(x) \) |
$ \frac{d}{dx} = f'(g(x)) \cdot g'(x) $
Let’s explore the derivatives of basic functions through practical examples to help solidify these concepts.
Derivatives of Constant Functions
We’ll begin with the simplest type of function: a constant. The derivative of any constant is always zero because a constant doesn’t vary, so its rate of change with respect to any variable is nil.
$$ \frac{d}{dx} c = 0 $$
Example. Consider the horizontal line represented by the function \( f(x) = 5 \). Its derivative is zero because, regardless of the value of \( x \), the function stays the same. $$ f'(x)= \frac{d \ 5}{dx} = 0 $$
Derivative of the Identity Function
The identity function simply returns the input variable \( x \). Its derivative is always 1, as it increases at a constant rate.
$$ \frac{d}{dx} x = 1 $$
Example. Take \( f(x) = 5x \). Here we’re differentiating a constant (5) multiplied by the identity function \( x^1 \). $$ f'(x) = \frac{d \ 5x}{dx} $$ According to the product rule, constants can be factored out of the derivative. $$ f'(x) = 5 \cdot \frac{d \ x}{dx} $$ Applying the power rule \( n \cdot x^{n-1} \) to differentiate \( x^1 \): $$ f'(x) = 5 \cdot [ 1 \cdot x^{1-1} ] = 5 \cdot [ 1 \cdot x^0 ] = 5 \cdot [ 1 \cdot 1 ] = 5 $$ Thus, the derivative of \( f(x)=5x \) is \( f'(x)=5 \).
Derivative of a Power Function
For power functions, we apply the power rule:
$$ \frac{d}{dx} x^n = n \cdot x^{n-1} $$
Example. Let’s look at \( f(x) = x^3 \). Its derivative is \( f'(x) = 3x^2 \). $$ f'(x) = 3 \cdot x^{3-1} = 3 x^2 $$
Derivative of Trigonometric Functions
Trigonometric functions each have their own specific derivatives:
- Derivative of sine $$ \frac{d}{dx} \sin(x) = \cos(x) $$
- Derivative of cosine $$ \frac{d}{dx} \cos(x) = - \sin(x) $$
- Derivative of tangent $$ \frac{d}{dx} \tan(x) = \frac{1}{ \cos^2(x) } = 1+ \tan^2 x = \sec^2(x) $$
- Derivative of cotangent $$ \frac{d}{dx} \cot(x) = - \frac{1}{ \sin^2(x) } = - (1 + \cot^2(x) ) $$
- Derivative of secant $$ \frac{d}{dx} \sec(x) = \frac{ \sin(x) }{ \cos^2(x)} = \sec(x) \cdot \tan(x) $$
- Derivative of cosecant $$ \frac{d}{dx} \csc(x) = - \frac{ \cos(x) }{ \sin^2(x)} = - \csc(x) \cdot \cot(x) $$
- Derivative of arcsine $$ \frac{d}{dx} \arcsin(x) = \frac{1}{ \sqrt{1-x^2} } $$
- Derivative of arccosine $$ \frac{d}{dx} \arccos(x) = - \frac{1}{ \sqrt{1-x^2} } $$
- Derivative of arctangent $$ \frac{d}{dx} \arctan(x) = \frac{1}{ 1+x^2 } $$
Derivative of Exponential and Logarithmic Functions
Exponential and logarithmic functions play a fundamental role in calculus as well:
- Derivative of the exponential function $$ \frac{d}{dx} e^x = e^x $$ $$ \frac{d}{dx} a^x = a^x \cdot \log a $$
- Derivative of the logarithm $$ \frac{d}{dx} \log_b(x) = \frac{1}{x} \log_b e $$
Derivative of a Composite Function
For composite functions, we use what’s called the chain rule.
$$ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) $$
One of the most frequent mistakes for beginners is forgetting that many functions are actually compositions of other functions.
For example, the function \( e^{2x} \) is a composition of the exponential function \( f = e^g \) and the linear function \( g = 2x \).
Example. A classic example of a composite function is an exponential function like \( f(x) = e^{2x} \). The derivative is \( f'(x) = 2e^{2x} \), because \( g(x) = 2x \) and \( g'(x) = 2 \).
This technique is known as the chain rule, and it can be applied even when multiple functions are nested, as in \( f(g(h(x))) \).
The idea is always the same: you differentiate step by step, moving from the outermost function inward.
\[ \frac{d}{dx} f[g(h(x))] = f'[g(h(x))] \cdot g'(h(x)) \cdot h'(x) \]
Committing these fundamental derivatives to memory is the foundation of differential calculus. From there, practice is the key to learning how to apply them effectively.
And so on.
