25. SLPOE OF NORMAL dyl dx =0 ture or false
Answers
Answer:
. True or False, and explain:
(a) The derivative of a polynomial is a polynomial.
True. A polynomial is a function of the form a0+a1x+a2x
2+. . . anx
n,
and its derivative will also have integer powers of x by the Power Rule,
nxn−1
.
(b) If f is differentiable, then d
dx
p
f(x) = f
0
(x)
2
√
f(x)
True:
d
dx
p
f(x) = d
dx (f(x))1/2 =
1
2
(f(x))−1/2
f
0
(x) = f
0
(x)
2
p
f(x)
(c) The derivative of y = sec−1
(x) is the derivative of y = cos(x).
False. The notation, sec−1
(x) is for the inverse secant function, which
is not the reciprocal of the secant.
For extra practice, to get the formula for the derivative of y =
sec−1
(x):
sec(y) = x
From this, draw a right triangle with one acute angle labelled y, the
hypotenuse x and the adjacent length x. This gives the length of the
side opposite: √
x
2 − 1. Now differentiate:
sec(y) tan(y)
dy
dx = 1
From the triangle, sec(y) = x and tan(y) = √
x
2 − 1, so:
dy
dx =
1
x
√
x
2 − 1
(d) d
dx (10x
) = x10x−1
False. The Power Rule can only be used for x
n, not a
x
. The derivative
is 10x
ln(10).
(e) If y = ln |x|, then y
0 =
1
x
.
TRUE. To see this, re-write the function:
y =
ln(x) if x > 0
ln(−x) if x < 0
⇒ y
0 =
1
x
if x > 0
1
−x
· (−1) if x < 0
From which we see that y
0 =
1
x
, x 6= 0.
(f) The equation of the tangent line to y = x
2 at (1, 1) is:
y − 1 = 2x(
Step-by-step explanation: