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(6) Compute the derivative of x4√ x

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Problem set 13: derivatives.

(1) Compute the derivative of the function x 7→ 8x8+ 3x2− 1.

(2) Give an example of a function that is not constant and whose derivative vanishes everywhere in the domain.

(3) Compute the derivative of the function x2+√

x on (0, ∞).

(4) Is |x| defined on R differentiable at 0?

(5) Let f : R → R be a function. We say that its symmetric derivative exists at some point x if the following limit exists:

h→0lim

f (x + h) − f (x − h)

2h .

Show that if the derivative exists at a point x, then so does the symmetric derivative, and the two agree at this point.

(6) Compute the derivative of x4√ x.

(7) Let f (x) = x3. Determine the sets where f0(x) = 0, f0(x) > 0 and f0(x) < 0.

(8) Calculate the derivative of

x3− 7x√5 x2 2√

x defined on (0, ∞).

(9) What is the derivative of

3

q x4

x on (0, ∞)?

(10) Compute the derivative of x5cos(x).

(11) Compute the derivative of the function x 7→ 1 − x2

7x2+ 9. (12) What is the derivative of

y = (7x5+ 6x3− x2− 1)3? (13) What is the derivative of sin2x + cos2x?

(14) Compute the derivative of sin(4x2).

(15) Calculate the derivative of sin5x.

(16) Determine the derivative of e−2x.

(17) Compute the derivative of tan4(3x) where the function is defined.

(18) What is the derivative of xx where it exists?

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