Mathfolis

Exponential & Log Derivatives

Unit 2 · Differentiation — Rules

Exponential and Logarithmic Derivatives

eˣ is its own derivative — a unique and fundamental property of the natural exponential function. The natural log derivative 1/x follows directly. For other bases, the chain of natural log carries through.

Essential Exponential & Log Derivatives

ddx[ex]=exddx[ln⁡x]=1x\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}

General Base Formulas

ddx[ax]=axln⁡addx[log⁡ax]=1xln⁡a\frac{d}{dx}[a^x] = a^x \ln a \qquad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}
AP Tip: eˣ is the only function that equals its own derivative. For any other base aˣ, a factor of ln a appears.
Type 1

Basic eˣ and ln x Derivatives

Differentiate expressions involving eˣ and ln x using the standard rules.

Example 1

Find f′(x).

f(x)=5ex−3ln⁡x+7f(x) = 5e^x - 3\ln x + 7
Example 2

Find the equation of the tangent line at x = 1.

y=ex+ln⁡xy = e^x + \ln x

Practice more of this type— AI-generated · infinite problems

Generate Problems →
Type 2

General Base — aˣ and log_a x

Apply the general base formulas when the base is a positive constant other than e.

Example 3

Find the derivative and evaluate at x = 0.

ddx[4x]\frac{d}{dx}[4^x]
Example 4

Find the derivative.

ddx[log⁡5x]\frac{d}{dx}[\log_5 x]

Practice more of this type— AI-generated · infinite problems

Generate Problems →
← Trig DerivativesImplicit Differentiation →