Composite Exponential Functions
When the exponent is a function of , the exponential is the outside function and the exponent is the inside function.
Differentiate the exponent
The exponential is the outside function; its exponent is the inside function. Both layers change.
A composite has layers
x
↓
inside: u = g(x)
rate g′(x)
↓
outside: y = f(u)
rate f′(u)
dy/dx = f′(g(x)) × g′(x)
Key idea
If , then .
Worked example
Differentiate a composite exponential
Combine rules
Worked example
A polynomial multiplied by an exponential
The chain rule differentiates . The product rule connects it to . One question may need both rules.
Use positivity
In , the factor is positive, so the stationary values come only from or .
Your turnIndependent check [5 marks]
Differentiate and find its stationary value of .
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Answer