3. Techniques of Differentiation

Given the initial-value problem: $\frac{d^{2}x}{d{t}^2}+c^{2}x=F_0\sin (ct)$, $x\left(0\right)=0$, $x^{\prime }\left(0\right)=0$, what is the general solution for $x\left(t\right)$?

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x³ - 3 (x² + π²)

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 4x²+1; a= 2,4

Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at

a. x=1

Derivatives Find and simplify the derivative of the following functions.

g(t) = 3t² + 6/t⁷

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.

<IMAGE>

G'(2)

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(x) = (√x+1)(√x-1)

The following limits represent f'(a) for some function f and some real number a.

Find a possible function f and number a.

lim x🠂0 e^x-1 / x

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

Find the indicated derivative.

$d/dt(-5t)$

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

y = x⁵

Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

r(t) = (e^2t + 3e^t + 2) / (e^t + 2)

Find the indicated derivative.

$d/dx\(\left\)(\(\pi\]\right\))$

Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

y = (x² - 2ax + a²) / (x - a); a is a constant.

Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).