What is the derivative of e^(3x)?
- e^(3x)
- 3e^(3x)
- 3x · e^(3x - 1)
What is the derivative of e^(3x)?
- e^(3x)
- 3x times e^(3x - 1)
- 3e^(3x)
What is the derivative of ln(5x)?
The derivative of e^x is e^x.
Circle one: True False
Differentiate ln(x^3).
Using the tangent line of sin x at x = 0, estimate sin(0.1). Give a decimal.
Answer: ______________
Differentiate ln(5x).
Differentiate cos(2x).
- -sin(2x)
- -2 sin(2x)
- 2 sin(2x)
Differentiate 2^x.
- x · 2^(x - 1)
- 2^x · ln 2
- 2^x
What is the derivative of 2^x?
- x times 2^(x - 1)
- 2^x ln 2
- 2^x
Derivatives of e, logs and trig W1-mt_695WPZKADT-s1
- 3e^(3x) · The outer function is e^u and the inner function 3x contributes a factor of 3.
- 3e^(3x) · Keep e^(3x) from the outer rule, then multiply by the inner derivative 3.
- 1/x · Split into ln 5 + ln x. The constant vanishes and ln x gives 1/x.
- True · That is what makes e special: the ln factor equals 1 and disappears.
- 3 / x · Chain rule: 1 over the inside, times the derivative of the inside, simplifies to 3/x.
- 0.1 · Near 0, sin x behaves like its tangent line y = x because the slope there is cos 0 = 1.
- 1 / x · Split ln(5x) into ln 5 + ln x; the constant drops out and the derivative of ln x is 1/x.
- -2 sin(2x) · Chain rule: the derivative of cos is -sin, and the inner 2x contributes a factor of 2.
- 2^x · ln 2 · For any base, the derivative of a^x is a^x ln a, so 2^x gains a factor of ln 2.
- 2^x ln 2 · Every base a keeps ln a: rewrite as e^(x ln 2) and the chain rule adds ln 2.