# Calculus Test - 10 Basic Problems

I need the full calculations as well as the answers. If I can see how you got from A to B to C I can figure it out, just the answers wouldn't help.10 basic problems.

Find the First Derivative

K(x)= 1/ (x^4)-(x^2)+1

G(t)= square root of 6t+5

G(x)= 6/[(3x^2)-1]^4

F(r)=[(r^2)-(r^-2)]^-2

Find the Limit

lim x^2/sinx

x->0

lim [(x^2)+sin^2x]/4x^2

x->0

Find the First Derivative

G(r)= square root of t+cos^2r

F(x)=sin^2(4x^3)

H(t)=(1+sin3t)^3

F(x)=x^2cot2x

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#### Solution Preview

Find the First Derivative

K(x)= 1/ (x^4)-(x^2)+1

K(x) = 1/(x^4 - x^2 + 1) = (x^4 - x^2 + 1)^-1

Let x^4 - x^2 + 1 = u

K(u) = u^-1

dK/du = -u^-2

du/dx = 4x^3 - 2x

dK/dx = dK/du * du/dx = u^-2 * (4x^3 - 2x) = 2x(2x^2 - 1)/(u^2) = 2x(2x^2 - 1)/ (x^4 - x^2 + 1)^2

G(t)= square root of 6t+5

G(t) = Ö(6t + 5)

Let 6t + 5 = u

G(u) = Öu

dG/du = 1/2Öu

du/dt = 6

dG/dt = dG/du * du/dt = (1/2Öu)6 = 3/Öu = 3/Ö(6t + 5)

G(x)= 6/[(3x^2)-1]^4

G(x) = 6(3x^2 - 1)^-4

Let 3x^2 - 1 = u

G(u) = 6u^-4

dG/du = 6(-4) u^-5 = -24/(u^5)

du/dx = 6x

dG/dx = ...

#### Solution Summary

The expert examines multiple derivatives and trigonometry functions. Complete, Neat and Step-by-step Solutions are provided in the attached file.