3) DIFFERENTIATION OF COMPOSITE FUNCTIONS, EQUATIONS OF
TANGENT AND NORMAL
APPLIED MATHEMATICS (FAPPZ)
1. Differentiation of composite functions
Compute the first derivative f 0 (x) of the function y = f (x).
Basic.
1) y =
√
1 − x2
2
4) y = (3x + 2x)
2) y =
5
√
3
a + bx3
5) y = arctg
3) y = arcsin 2x
6) y = 5e−x
1
x
2
From examinations.
7) y = arctg2
1
x
8) y = ln
10) y = x arccos x −
√
q
x−1
x+1
9) y = arcsin 1−x
1+x
√
√
11) y = 4 4 x − 4 arctg 4 x
1 − x2
12) y =
1 + cos 2x
1 − cos 2x
Advanced.
13) y = −
√
arcsin x
2
+ ln 1− x1−x
x
√
2
2
+a +x
14) y = ln √xx2 +a
2 −x
15) y = 2 arctg x −
ln(x2 + 1)
x
Compute the second derivative f 00 (x) of the function y = f (x).
16) y =
2x + 3
5x − 1
17) y =
1 − ln x
1 + ln x
18) y =
3 + e2x
5 − e2x
Results.
−x
1 − x2
1)
√
4)
5(3x2 + 2x)4 (6x + 2)
7)
−
2)
2 arctg x1
1 + x2
10)
arccos x
13)
arcsin x
x2
16)
170
(5x − 1)3
bx2
p
3
(a + bx3 )2
−1
5)
1 + x2
1
8)
2
x −1
1
√
11) √
4
x(1 + x)
2
14) √
2
x + a2
2(3 + ln x)
17)
x2 (1 + ln x)3
2
1 − 4x2
3)
√
6)
− 10xe−x
9)
√
12)
15)
18)
2
−1
x(1 + x)
cos x
−2 3
sin x
ln(x2 + 1)
x2
32e2x (5 + e2x)
(5 − e2x )3
2. Equations of tangent and normal
Basic. 1) Find an equation of the tangent line to the graph of the function f : y = x2 − 3x + 1
at the point of its graph T = [2, yT ].
x+1
2) Find an equation of the normal line to the graph of the function f : y =
at the
x+3
point of its graph T = [−2, yT ].
c Petr Gurka
(updated October 2, 2011).
1
2
DIFFERENTIATION, TANGENT, NORMAL
From examinations. 3) Find equations of the tangent and the normal to the graph of the
x
function f : y =
at the point of its graph T = [π, yT ].
1 − cos x
4) Find equations of the tangent and the normal to the graph of the function f : y =
1+x
at the point of its graph T = [0, yT ].
arccotg 1−x
Advanced. 5) Find an equation of the tangent to the curve y = x2 − 1 so that the tangent is
parallel with line p : 2x − y + 3 = 0.
6) Find an equation of the normal to the curve y = x2 so that the normal is parallel with
line p : 2x − 6y + 5 = 0.
7) Find an equation of the tangent to the curve y = x2 so that the tangent is perpendicular
to line p : 2x − 6y + 12 = 0.
1
parallel with x-axis?
8) In which point is the curve y = 2
x − 4x + 5
x+1
9) Find an equation of the normal to the curve y = ln xe2 +1 so that the normal is parallel
with line p : 2x + 4y + 5 = 0.
Results.
2) n : y = − 21 x − 2
1) t : y = x − 3
3) t : y = 21 x,
n : y = −2x + 25 π
6) n : y = 13 x +
5) t : y = 2x − 2
7) t : y = −3x −
4) t : y = −x + π4 ,
9
4
9) n : x + 2y + 1 + ln 4 = 0
8) A = [2, 1]
5
6
n:y =x+
π
4