A Generalized Power Rule Calculator is a mathematical tool designed to compute derivatives of functions that involve variables raised to constant or variable powers. This calculator falls under the category of "Mathematics and Calculus Tools". It simplifies the differentiation process by applying the generalized power rule formula directly, saving time and reducing the risk of manual errors. It can handle both constant exponents and variable exponents, making it useful for students, educators, engineers, and professionals dealing with complex mathematical expressions.
The calculator works by taking the function, identifying the base u(x), the exponent n or v(x), and applying the correct differentiation formula. It can also handle scenarios where logarithmic differentiation is necessary, especially when the exponent itself depends on x.
formula
For a function:
f(x) = [u(x)]^n
Derivative formula:
f'(x) = n × [u(x)]^(n-1) × u'(x)
Where:
u'(x) = derivative of u(x) with respect to x
If n is not constant but a function of x (n = v(x)), use the generalized logarithmic differentiation:
f'(x) = [u(x)]^(v(x)) × [ v'(x) × ln(u(x)) + v(x) × (u'(x) / u(x)) ]
Sub-formulas:
u'(x) = du/dx
v'(x) = dv/dx
ln(u(x)) = natural logarithm of u(x)
Commonly Searched Generalized Power Rule Terms Table
| Term | Meaning | Example |
|---|---|---|
| u(x) | The base function inside the power | 3x² + 5 |
| n | Constant exponent | 4 in (x³)^4 |
| v(x) | Variable exponent function | sin(x) in (x²)^(sin(x)) |
| u'(x) | Derivative of u(x) | d/dx(3x² + 5) = 6x |
| v'(x) | Derivative of v(x) | d/dx(sin(x)) = cos(x) |
| ln(u(x)) | Natural log of the base | ln(3x² + 5) |
| Generalized Power Rule | Formula for derivative of u(x)^n or u(x)^(v(x)) | See formula above |
Example
Let u(x) = x² + 1 and n = 5.
Step 1: Identify u'(x)
u'(x) = d/dx(x² + 1) = 2x
Step 2: Apply formula for constant exponent
f'(x) = n × [u(x)]^(n-1) × u'(x)
f'(x) = 5 × (x² + 1)^(5-1) × 2x
f'(x) = 10x × (x² + 1)^4
Now, if n is variable, let v(x) = sin(x), u(x) = x² + 1.
u'(x) = 2x
v'(x) = cos(x)
f'(x) = (x² + 1)^(sin(x)) × [cos(x) × ln(x² + 1) + sin(x) × (2x / (x² + 1))]
Most Common FAQs
The generalized power rule is used to find the derivative of a function where the base and/or exponent may be a function of x.
The basic power rule applies only when the base is x and the exponent is a constant. The generalized power rule works for any function raised to a constant or variable exponent.
Yes, it can differentiate functions with trigonometric exponents like sin(x), cos(x), or other functional powers using logarithmic differentiation.