Derivatives of Standard Functions
Trigonometric Functions
The derivatives of the basic trigonometric functions are fundamental in calculus:
- $\frac{d}{dx}(\sin x) = \cos x$
- $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(\tan x) = \sec^2 x$
Let's differentiate $f(x) = \sin(2x)$:
Using the chain rule (which we'll explore later), we get: $f'(x) = \cos(2x) \cdot 2 = 2\cos(2x)$
NoteThe derivatives of inverse trigonometric functions, while not explicitly mentioned in the syllabus, are often useful in more advanced problems:
- $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$
Exponential and Logarithmic Functions
The exponential and natural logarithm functions have particularly elegant derivatives:
- $\frac{d}{dx}(e^x) = e^x$
- $\frac{d}{dx}(\ln x) = \frac{1}{x}$
Differentiate $g(x) = e^{3x}$:
Again, using the chain rule: $g'(x) = e^{3x} \cdot 3 = 3e^{3x}$
Power Function
For the general power function:
- $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a rational number
This rule is incredibly versatile, applying to integer, fractional, and negative exponents.
ExampleDifferentiate $h(x) = x^{-1/2}$:
$h'(x) = -\frac{1}{2}x^{-3/2}$
Common MistakeStudents often forget that this rule applies to negative and fractional exponents. Remember, $x^{-1} = \frac{1}{x}$ and $x^{1/2} = \sqrt{x}$.
Chain Rule
The chain rule is a powerful tool for differentiating composite functions. If $y = f(u)$ and $u = g(x)$, then:
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
In Leibniz notation, this is often written as:
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$
ExampleDifferentiate $y = \sin(x^2)$:
Let $u = x^2$, then $y = \sin(u)$
$\frac{dy}{du} = \cos(u)$ $\frac{du}{dx} = 2x$
Therefore, $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$
TipWhen applying the chain rule, it can be helpful to think of it as "inside function" times "outside function". The derivative of the inside function is multiplied by the derivative of the outside function composed with the inside function.
Product Rule
The product rule is used to differentiate the product of two functions. If $y = u(x)v(x)$, then:
