Calculate derivatives with the generalized power rule for u(x)^n or u(x)^v(x), then evaluate f(x), f′(x), and the tangent line at x.

Differentiate a power of a function.

Use * for multiplication, for example 3*x^2. Trigonometric angles use radians.

Leave blank for the symbolic derivative.

Related Calculators

Generalized Power Rule Formula

The following formula is used to calculate the derivative using the Generalized Power Rule.

d / dx [uⁿ] = n × u⁽ⁿ⁻¹⁾ × u'

Variables:

  • d/dx [u^n] is the derivative of the function u^n with respect to x
  • u is a differentiable function of x
  • n is a constant real number (the power to which u is raised)
  • u’ is the derivative of u with respect to x

To calculate the derivative using the Generalized Power Rule, first identify the function u and its power n. Then, calculate the derivative of u with respect to x (u’). Multiply the power n by the function u raised to the power of n-1, and then multiply this result by u’. This gives the derivative of the function u^n with respect to x.

What is a Generalized Power Rule?

The Generalized Power Rule is a formula used in calculus to differentiate functions that are in the form of a power of a function. It combines the basic power rule with the chain rule when the exponent n is constant. For a variable exponent v(x), use logarithmic differentiation: d/dx [u(x)^v(x)] = u(x)^v(x)*[v′(x)*ln(u(x)) + v(x)*u′(x)/u(x)], on a domain where u(x) is positive. The rule states that the derivative of u^n, where u is a differentiable function of x and n is a real number, is n*u^(n-1)*u’, where u’ is the derivative of u with respect to x.

How to Calculate Generalized Power Rule?

The following steps outline how to calculate the derivative of a function using the Generalized Power Rule.

  1. First, identify the function u and the power n.
  2. Next, differentiate the function u with respect to x to find u’.
  3. Then, apply the Generalized Power Rule formula: d/dx [u^n] = n*u^(n-1)*u’.
  4. Finally, substitute the values of u, n, and u’ into the formula and simplify the expression.
  5. After simplifying, check your answer with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

u = 3x^2 (enter 3*x^2 in the calculator)

n = 4

u’ = 6x