If y = f(g(x)), then dy/dg(x) is equal to:
A
f'(x)
B
f'(g(x))
C
g'(x)
D
f(g'(x))
Analysis & Theory
Using chain rule: dy/dx = f'(g(x))·g'(x), so dy/dg(x) = f'(g(x))
If y = sin(x²), then dy/d(x²) = ?
A
cos(x²)
B
2x·cos(x²)
C
cos(x²)/2x
D
sin(x²)
Analysis & Theory
Let u = x². Then y = sin(u), so dy/du = cos(u) = cos(x²)
If y = ln(x² + 1), find dy/d(x² + 1):
A
1 / (x² + 1)
B
2x / (x² + 1)
C
ln(x² + 1)
D
1 / x
Analysis & Theory
Let u = x² + 1 ⇒ y = ln(u), so dy/du = 1/u
If f(x) = e^(3x), then d/d(3x)[f(x)] = ?
A
3e^(3x)
B
e^(3x)
C
1/3 · e^(3x)
D
ln(3x)
Analysis & Theory
Let u = 3x ⇒ f = e^u ⇒ d(f)/d(u) = e^u = e^(3x)
If y = tan(x³), then dy/d(x³) is:
A
sec²(x³)
B
3x²·sec²(x³)
C
tan(x³)
D
1 + tan²(x³)
Analysis & Theory
Let u = x³ ⇒ y = tan(u) ⇒ dy/du = sec²(u) = sec²(x³)
Which rule is used to differentiate one function with respect to another?
A
Product rule
B
Quotient rule
C
Chain rule
D
Power rule
Analysis & Theory
The chain rule is used when differentiating a function of another function.
If y = (x² + 1)³, then dy/d(x² + 1) is:
A
3(x² + 1)²
B
6x(x² + 1)
C
3x²(x² + 1)
D
None of these
Analysis & Theory
Let u = x² + 1 ⇒ y = u³ ⇒ dy/du = 3u² = 3(x² + 1)²
If y = sin²(x), then dy/d(sin(x)) is:
A
2sin(x)
B
cos(x)
C
2sin(x)cos(x)
D
2sin(x)
Analysis & Theory
Let u = sin(x) ⇒ y = u² ⇒ dy/du = 2u = 2sin(x)
If y = f(x), and x = g(t), then dy/dt = ?
A
f'(x)
B
f'(x)·g'(t)
C
g'(t)
D
f(g(t))
Analysis & Theory
Using chain rule: dy/dt = dy/dx · dx/dt = f'(x)·g'(t)
If y = √(x² + 1), then dy/d(x²) = ?
A
1 / √(x² + 1)
B
x / √(x² + 1)
C
1 / (2√(x² + 1))
D
x² / √(x² + 1)
Analysis & Theory
Let u = x² ⇒ y = √(u + 1) ⇒ dy/du = 1 / (2√(u + 1))