QCAA Mathematical Methods Differentiation of trigonometric functions and differentiation rules

This incorrect value is half of the correct answer, which may result from a calculation error during the multiplication of fractions or evaluating the trigonometric ratio.

This answer results from evaluating $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ instead of $\cos\left(\frac{\pi}{3}\right)$ in the derivative, or incorrectly assuming the derivative of sine is sine.

This option is incorrect and likely results from misapplying the chain rule or arithmetic errors when combining the constants.

This option implies a slope of $e$, but the derivative at $t=1$ is $f'(1) = e^1 + 1\cdot e^1 = 2e$.

This option uses the incorrect slope $e$ instead of $2e$, likely failing to apply the product rule correctly when finding the derivative.

While the slope $2e$ is correct, the y-intercept is wrong. Expanding $y - e = 2e(t - 1)$ results in a constant term of $-e$, not $-2e^2 + 1$.

This is the minimum value of the function on the interval, which occurs at the critical point $x = 1$ since $h(1) = -e$.

This is the $x$-value of the critical point (found by setting $h'(x) = (x-1)e^x = 0$), not the maximum value of the function itself.

This is the $x$-value at which the maximum occurs, but the question asks for the maximum value of the function, which is $h(2) = 0$.

This value does not correspond to the function evaluated at any critical point or endpoint within the given interval $[0, 2]$.

This is incorrect. It confuses the derivative of the exponent with the exponent itself. The term $e^{\sin(x)}$ should remain, multiplied by the derivative of the sine function.

This is incorrect because it ignores the chain rule. While the derivative of $e^x$ is $e^x$, the derivative of a composite function $e^{u(x)}$ requires multiplying by $u'(x)$.

This is incorrect. You cannot simply differentiate the exponent in place; the chain rule requires preserving the original exponential term $e^{\sin(x)}$ and multiplying by the derivative of the exponent.

This option is incorrect because while $f'(-1)=0$, the second derivative $f''(-1) = e^{-1}(-1+2) = e^{-1}$ is positive, not negative.

This option is incorrect because the first derivative evaluates to zero at $x=-1$, not a negative value, and the second derivative is positive.

This option is incorrect because the first derivative $f'(-1)$ equals $0$, not a value less than $0$.

The number of local minima depends on the value of the positive integer $a$ and is not a constant $2$.

This would only be correct if $a=2$, but the number of local minima changes depending on the specific value of $a$.

The number of periods in the interval is calculated by dividing the interval length by the period, which yields $a^2$, not $a$.

While the interval length is $2a\pi$, the period of the function is $2\pi/a$. Dividing these gives $a^2$ local minima, not $2a$.

This option is incorrect because it misses the derivative of the inner function $g$. By the chain rule, you must multiply by $g'(x^2)$.

This option is incorrect because it multiplies by the original function $g(x^2)$ instead of its derivative $g'(x^2)$.

This option is incorrect because it evaluates the derivative $g'$ at $2x$ instead of $x^2$. The chain rule requires evaluating the outer derivative at the inner function.

This option is incorrect because it changes the exponent of $e$ to $g(2x)$. The outer function $e^{g(x^2)}$ should remain unchanged when applying the first step of the chain rule.

This option has the wrong sign. The derivative of $\cos(3x)$ is $-3\sin(3x)$, so the second term in the product rule expansion should be negative.

This option ignores the product rule. It incorrectly treats $2x$ as a constant multiplier or only differentiates the cosine term while discarding the first part of the product rule expansion.

This option fails to apply both the product rule and the chain rule correctly. It misses the derivative of the $2x$ term and the inner derivative of $3x$.