Derivative rules for composite and implicit functions - One Line Questions
1.
What is the derivative of f(x) = cot(x^2)? —
-2x csc^2(x^2)
2.
If y = cos(x^2 + 1), find dy/dx. —
-2x sin(x^2 + 1)
3.
If y = e^(cos(x)), find dy/dx. —
-e^(cos(x)) sin(x)
4.
What is the derivative of f(x) = cos(e^x)? —
-e^x sin(e^x)
5.
Find the derivative of h(x) = cos(sin(x)). —
-sin(sin(x)) cos(x)
6.
If x^2 + y^2 = 25, find dy/dx. —
-x/y
7.
Find the derivative of y = e^(x^2 + x). —
(2x + 1) e^(x^2 + x)
8.
If x^3 + y^3 = 3xy, find dy/dx. —
(y - x^2) / (y^2 - x)
9.
What is the derivative of y = arctan(x/a)? —
a / (a^2 + x^2)
10.
If y = sqrt(1 - x^2), find dy/dx. —
-x / sqrt(1 - x^2)
11.
If y = log_a(x), find dy/dx. —
1 / (x ln(a))
12.
What is the derivative of f(x) = arcsin(x/a)? —
1 / sqrt(a^2 - x^2)
13.
What is the derivative of f(x) = arcsin(2x)? —
2 / sqrt(1 - 4x^2)
14.
If y = (ln(x))^2, find dy/dx. —
2 ln(x) / x
15.
What is the derivative of f(x) = log(1 + x^2) with respect to x? —
2x / (1 + x^2)
16.
What is the derivative of f(x) = tan^{-1}(x^2)? —
2x / (1 + x^4)
17.
Find the derivative of y = ln(x^2 + 1). —
2x / (x^2 + 1)
18.
What is the derivative of f(x) = sin^{-1}(x^2)? —
2x / sqrt(1 - x^4)
19.
If y = sin(x^2) + cos(x^2), find dy/dx. —
2x cos(x^2) - 2x sin(x^2)
20.
Find the derivative of y = x^2 * cos(x). —
2x cos(x) - x^2 sin(x)
21.
If y = x^2 * e^(sin(x)), find dy/dx. —
2x e^(sin(x)) + x^2 e^(sin(x)) cos(x)
22.
If y = x^2 * ln(x), find dy/dx. —
2x ln(x) + x
23.
Find the derivative of y = x^2 * sin(x). —
2x sin(x) + x^2 cos(x)
24.
What is the derivative of f(x) = (x^2 + 1)^3? —
6x(x^2 + 1)^2
25.
If y = x^3 * e^(2x), find dy/dx. —
3x^2 e^(2x) + 2x^3 e^(2x)
26.
Find the derivative of y = e^(x^3). —
3x^2 e^(x^3)
27.
What is the derivative of f(x) = sec(x^3)? —
3x^2 sec(x^3) tan(x^3)
28.
If y = (3x + 2)^4, find dy/dx. —
12(3x + 2)^3
29.
If y = sqrt(ax + b), find dy/dx. —
a / (2 * sqrt(ax + b))
30.
Find the derivative of y = sin(ax + b). —
a cos(ax + b)
31.
Differentiate f(x) = a^(g(x)) where g(x) is a function of x. —
a^(g(x)) g'(x) ln(a)
32.
If y = x * arctan(x), find dy/dx. —
arctan(x) + x / (1 + x^2)
33.
Find the derivative of y = sin(x)cos(x). —
cos(2x)
34.
What is the derivative of f(x) = sin(x^2) with respect to x? —
2x cos(x^2)
35.
What is the derivative of f(x) = sqrt(sin(x))? —
cos(x) / (2 * sqrt(sin(x)))
36.
Find the derivative of y = ln(sin(x)). —
cot(x)
37.
If y = e^(sin(x)), find dy/dx. —
e^(sin(x)) cos(x)
38.
If y = e^(tan^{-1}(x)), find dy/dx. —
e^(tan^{-1}(x)) / (1 + x^2)
39.
Given f(x) = e^(tan(x)), what is f'(x)? —
e^(tan(x)) sec^2(x)
40.
Find the derivative of y = sin^{-1}(e^x). —
e^x / sqrt(1 - e^(2x))
41.
Given y = e^(kx), find dy/dx. —
k e^(kx)
42.
Find the derivative of y = (sin(x))^n. —
n (sin(x))^(n-1) cos(x)
43.
If y = x^n, find dy/dx. —
nx^(n-1)
44.
Find the derivative of y = x^n * e^x. —
nx^(n-1)e^x + x^n e^x
45.
If y = tan(3x), what is dy/dx? —
3 sec^2(3x)
46.
What is the derivative of f(x) = tan(e^x)? —
sec^2(e^x) * e^x
47.
If y = tan(x^2 + x), find dy/dx. —
sec^2(x^2 + x) * (2x + 1)
48.
What is the derivative of f(x) = ln(tan(x))? —
sec(x) csc(x)
49.
Find the derivative of y = x * sin^{-1}(x). —
sin^{-1}(x) + x / sqrt(1 - x^2)
50.
What is the derivative of g(x) = ln(cos(x))? —
-tan(x)