Differentiate logarithmic and exponential functions such as ln(x), ex, ax, and nested expressions using the chain rule.
Free online symbolic calculator with derivative rules and step-by-step explanations.
The most important rules used by this calculator are:
\[ \frac{d}{dx}\ln(x)=\frac{1}{x} \]
\[ \frac{d}{dx}\ln(u)=\frac{u'}{u} \]
\[ \frac{d}{dx}e^x=e^x \]
\[ \frac{d}{dx}e^u=e^u u' \]
\[ \frac{d}{dx}a^x=a^x\ln(a) \]
\[ \frac{d}{dx}a^u=a^u\ln(a)u' \]
Logarithmic differentiation is a technique for differentiating functions that can be difficult to differentiate directly. Taking the natural logarithm can transform products into sums and powers into multiplication, making the derivative easier to calculate.
For example, if \(y=x^x\), taking logarithms gives \(\ln(y)=x\ln(x)\). Differentiating both sides allows the derivative of the original variable-exponent function to be found.
The technique is especially useful for functions involving products, quotients, variable exponents, and complicated powers.
When a logarithm or exponential contains another function, the chain rule is required.
For example:
Let \(u=x^2+1\). Then \(u'=2x\). Therefore:
\[ f'(x)=\frac{2x}{x^2+1} \]The same idea applies to exponential functions:
Since \(u=x^2\) and \(u'=2x\):
\[ f'(x)=2xe^{x^2} \]| Function | Derivative | Rule |
|---|---|---|
| \(\ln(x)\) | \(\frac{1}{x}\) | Natural logarithm rule |
| \(\ln(x^2+1)\) | \(\frac{2x}{x^2+1}\) | Logarithmic chain rule |
| \(e^x\) | \(e^x\) | Exponential rule |
| \(e^{2x}\) | \(2e^{2x}\) | Exponential chain rule |
| \(2^x\) | \(2^x\ln(2)\) | General exponential rule |
| \(5^{x^2}\) | \(5^{x^2}\ln(5)(2x)\) | Exponential chain rule |
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What is the derivative of ln(x)?
The derivative of the natural logarithm is \[ \frac{d}{dx}\ln(x)=\frac{1}{x}. \]
What is the derivative of e^x?
The exponential function \(e^x\) is its own derivative: \[ \frac{d}{dx}e^x=e^x. \]
What is the derivative of e^(u)?
If \(u\) is a function of x, the chain rule gives: \[ \frac{d}{dx}e^u=e^u u'. \]
What is the derivative of a^x?
For a positive constant \(a\): \[ \frac{d}{dx}a^x=a^x\ln(a). \]
When should I use logarithmic differentiation?
Logarithmic differentiation is particularly useful for functions containing variable exponents, complicated products, quotients, and powers where directly applying the ordinary power rule is difficult.