Fundamentals of Differentiation

INCORRECT. $x\sin(x)$

INCORRECT. $x^2\cos(x)$

CORRECT. $2x\sin(x) + x^2\cos(x)$

INCORRECT. $\cos(x) + 2x^3$

$f'(p) = ((p^3+p^3) \cdot (p+p)) + ((p^3-p^2) \cdot (p+p))$

$f'(p) = (p^3+p^3) + (p^3-p^2) \cdot p + (p+6) \cdot (3p^2-2p)$

$f'(p) = (p+6)^{(p^3+p^2)} \cdot ((p+6)^{(p^3-p^2)})^p$

$f'(p) = (p+6) \cdot (p^3+p^2) + (p-p) \cdot (p^3+p^2)$

$2x + e^x$

$x^2e^{x-1}$

$2xe^x + x^2e^x$

$2xe^x$

CORRECT. The rate of change of the product of $t$ and $e^{t}$

INCORRECT. The maximum value of the function at that point

INCORRECT. The accumulated area under the curve from zero to point $t$

INCORRECT. The y-intercept of the tangent line to the curve at that point

$g'(x) = 4\cos x - \sin x$

$g'(x) = \sin(4+x)$

$g'(x) = 4\sin x + 4x\cos x$

$g'(x) = \cos(4-x)$

$5(\sin(x) + x\cos(x))$

$5(\cos(x) + x\cos(x))$

$5(\sin(x) + x\sin(x))$

$5(\cos(x) + x\sin(x))$

Other techniques would result in an undefined derivative.

It avoids unnecessary algebraic complexity from expansion.

It's required since cosine cannot be differentiated alone.

It provides practice for integral calculus applications.

$f'(x) = 2xe^x + x^2e^x$

$f'(x) = 2xe^x - x^2e^{x-1}$

$f'(x) = e^{2x}$

$f'(x) = xe^{x+1}$

INCORRECT 2. $(x \sin(f(x)) - \cos(x))$

CORRECT. $(-x \cos(x) - \sin(x) + x \cos(x))$

INCORRECT 3. $(x \cos(f(x)) - \sin(x))$

INCORRECT 1. $(-\sin(x) + x \cos(x) - \cos(x))$

$8x^3 - 3x^2 - 4x - 1$

$8x^3 - 3x^2 + 4x - 1$

$8x^3 - 3x^2 + 4x + 1$

$8x^3 - 3x^2 + 4x$