How do you differentiate a function with an exponent?

Differentiating a function with an exponent depends on the form of the function. The approach differs if the exponent is a constant versus another function, or if the base is a variable or a constant. This guide will cover common scenarios and differentiation techniques.

Differentiating Power Functions (Constant Exponent)

When you have a function of the form f(x) = xⁿ, where n is a constant, you can use the power rule:

d/dx (xⁿ) = nx^(n-1)*

Example:

If f(x) = x³, then f'(x) = 3x².

Differentiating Exponential Functions (Constant Base)

When the base is a constant, such as f(x) = a^x, where a is a constant, the derivative is:

d/dx (a^x) = a^x ln(a)

Example:

If f(x) = 2^x, then f'(x) = 2^x ln(2).

A special case is when a = e (Euler's number):

d/dx (e^x) = e^x

Differentiating Functions with Variable Exponents

When both the base and the exponent are functions of x, such as f(x) = u(x)^v(x), you can use logarithmic differentiation or the chain rule.

Logarithmic Differentiation

1. Take the natural logarithm of both sides: ln(f(x)) = v(x) ln(u(x)).

2. Differentiate both sides with respect to x, using the product rule and chain rule:

   (f'(x) / f(x)) = v'(x) ln(u(x)) + v(x) (u'(x) / u(x))

3. Solve for f'(x):

   f'(x) = f(x) [v'(x) ln(u(x)) + v(x) (u'(x) / u(x))]

   f'(x) = u(x)^v(x) [v'(x) ln(u(x)) + v(x) (u'(x) / u(x))]

Example:

Let f(x) = x^x.

1. ln(f(x)) = x ln(x)
2. Differentiating both sides: (f'(x) / f(x)) = (1)ln(x) + x(1/x) = ln(x) + 1
3. Solving for f'(x): f'(x) = x^x (ln(x) + 1)

Chain Rule Approach

As mentioned in the reference, you can also think about it using the chain rule. Rewrite u(x)^v(x) as e^{v(x) ln(u(x))}. Then:

1. Differentiate the entire function with respect to e^{v(x) ln(u(x))}, which gives you e^{v(x) ln(u(x))}, or u(x)^v(x).
2. Multiply by the derivative of v(x) ln(u(x)) with respect to x, which requires the product rule: v'(x)ln(u(x)) + v(x) (u'(x)/u(x))*.
3. The final result is: u(x)^v(x) [v'(x) ln(u(x)) + v(x) (u'(x) / u(x))], which is the same result as obtained with logarithmic differentiation.

Examples and Practical Insights

• Example 1: Differentiate f(x) = (x² + 1)³

  • Using the chain rule: f'(x) = 3(x² + 1)² (2x) = 6x(x² + 1)²*

• Example 2: Differentiate f(x) = e^sin(x)

  • Using the chain rule: f'(x) = e^sin(x) cos(x) = cos(x)e^sin(x)*

• Practical Insight: When dealing with complex exponents, logarithmic differentiation often simplifies the process. Always remember to consider the base and exponent's form to choose the most efficient differentiation technique.

In summary, differentiating functions with exponents requires understanding the power rule, exponential function rules, the chain rule, and logarithmic differentiation. The choice of method depends on whether the exponent is a constant, a variable, or another function.