3. Techniques of Differentiation

Which of the following statements about differentiation is true?

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 $x\left(t\right)$?

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.

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G'(2)

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

(g^-1)'(7)

Find the indicated derivative.

$f(x)=3x-4$

$f^{\prime}(x)=$

Find the indicated derivative.

$d/dt(-5t)$

Find the indicated derivative.

$d/dt(-5x)$

Find the indicated derivative.

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

Find the indicated derivative.

$f(x)=6x^4-4^3$

$f^{\prime}\left(x\right)=$

Find the indicated derivative.

$d/dt(5-7t^{-3})$

Find the indicated derivative.

$y=\frac{5}{2}{x}^5+2x^3+6x-\frac{\surd 3}{4}$

$y^{\prime }=$

Find the indicated derivative.

$h\left(t\right)=\frac{1}{2}{t}^3+\frac{4}{t^2}+3\sqrt{t}$

$h^{\prime }\left(t\right)=$

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

a. d/dx (f(x)+2g(x)) |x=3

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (5f(x)+3g(x)) |x=1

Derivatives of even and odd functions Recall that f is even if f(−x) = f(x), for all x in the domain of f, and f is odd if f(−x) = −f(x) for all x in the domain of f.

b. If f is a differentiable, odd function on its domain, determine whether f' is even, odd, or neither.