By the chain rule, d/dx[e^(g(x))] = e^(g(x))·g'(x). Here g(x) = 3x², so g'(x) = 6x. Therefore d/dx[e^(3x²)] = 6x·e^(3x²). Forgetting the chain rule factor gives just e^(3x²) (option 0). Using 3 instead of 6 (from differentiating x² as x) gives option 1.
Question 2 True / False
The derivative of 2^x is the same as the derivative of e^x because both are exponential functions.
TTrue
FFalse
Answer: False
d/dx[e^x] = e^x, but d/dx[2^x] = 2^x·ln(2). Only the base e has the property that ln(e) = 1, making the correction factor disappear. For any other base b, the factor ln(b) appears. This is precisely what makes e the natural base for calculus.
Question 3 Short Answer
Explain why the power rule cannot be used to differentiate e^x, and state the correct derivative.
Think about your answer, then reveal below.
Model answer: The power rule d/dx[x^n] = n·x^(n-1) applies when the variable is the base and the exponent is a constant. In e^x, the variable x is the exponent and e is the constant base — the roles are reversed. The correct derivative is d/dx[e^x] = e^x.
Applying the power rule incorrectly would give x·e^(x-1), which is wrong. The confusion often arises because x^e (power function, e is constant) and e^x (exponential function, x is exponent) look similar but behave differently. x^e differentiates to e·x^(e-1); e^x differentiates to itself.