Fundamentals of Differentiation

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

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

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

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

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

$2x + e^x$

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

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

$2xe^x$

INCORRECT. $x\sin(x)$

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

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

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

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

No, the constant multiple rule is the most appropriate rule to use.

No rule can be applied to find the derivative of this function.

Yes, the product rule is the most appropriate rule to use.

No, the power rule is the most appropriate rule in to use.

INCORRECT 1. $((u)'(e) - y \cos(y) - 1)$

INCORRECT 3. $((u(e) \cos(y) - y - 1))$

CORRECT. $((u)'(e) - v'(e))$

INCORRECT 2. $((u'(e) - v'(e))$

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

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

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

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

INCORRECT. $\frac{dp}{dy}\cdot \frac{dq}{dy}$

CORRECT. $\frac{dp}{dy}\cdot q(y) + p(y)\cdot \frac{dq}{dy}$

INCORRECT. $(p+q)\left(\frac{dp}{dy}+\frac{dq}{dy}\right)$

INCORRECT. $p(y) + q(y)$

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

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

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

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