Differentiate sine, cosine and tangent functions and solve simple derivative conditions

Which first step is correct for the displayed function?

Change the positive integer frequency k

Which conclusion fits every setting?

Derivatives of trigonometric functions

Trigonometric derivatives use radian measure and the chain rule for inner expressions. After differentiating, exact trigonometric values and periodicity help solve any stated condition

1. 1Use radians Use radians when applying these derivative formulae. Trigonometric equations can be solved in either angle unit if it is used consistently \(\displaystyle 30^\circ=\frac{\pi}{6}\)
2. 2Recall the base derivatives Sine differentiates to cosine, cosine differentiates to negative sine and tangent differentiates to one over cosine squared \(\displaystyle \frac{d}{du}\sin u=\cos u,\quad \frac{d}{du}\cos u=-\sin u,\quad \frac{d}{du}\tan u=\frac{1}{\cos^2u}\)
3. 3Include the inner derivative For a trigonometric function of an inner expression, multiply by the derivative of that inner expression \(\displaystyle \frac{d}{dx}\sin(4x)=4\cos(4x)\)
4. 4Use identities and periods Use exact values, identities and the period of the function to find every required solution in the interval \(\displaystyle \sin^2u+\cos^2u=1,\quad \sin\left(u+2\pi\right)=\sin u\)

Given

\(\displaystyle u(x)=3x\)

Calculate the derivative of the given inner expression

Given

\(\displaystyle f'(0)=4(3)\cos(3(0))=12\cos(0)\)

Use the inner derivative to calculate the derivative at zero for the given function

Which differentiation line is correct?

Keep every circular chain factor

Angles are in radians. The cosine derivative contributes a minus sign, while the derivative of tangent is sec² of its complete argument. Exclude every pole of tangent from both the function and derivative

Construct the full expression

Construct the full derivative, state the excluded inputs using an integer k, and find its exact value at zero. Show the chain factor for every term; do not write only a gradient value

Task

\(\displaystyle g(x)=3\cos2x-4\sin(x/2)+2\tan3x\)

Find the derivative of the given function at zero

Review the whole expression

Each term needs its inner chain factor. Exclude every odd multiple of one sixth of pi from the function and derivative. Exact evaluation at zero provides a check but cannot replace the full rule

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression