Mixed trigonometric and polynomial functions, exact angles, product and quotient rules, second derivatives and tangents.
Example 1
f(x)=4sin(4x)+2x2, with x in radians. Which expression is f′(x)?
Show worked answer
Answer: 16cos(4x)+4x
- Differentiate the sum term by term.
- The chain rule gives (4sin(4x))′=16cos(4x).
- The power rule gives (2x2)′=4x, so f′(x)=16cos(4x)+4x.
Example 2
f(x)=4sin(4x), with x in radians. Find the exact value of f′(3π).
Show worked answer
Answer: −8
- Chain rule: f′(x)=16cos(4x).
- At x=3π, the angle 4x=34π and cosine is 2−1.
- Therefore f′(3π)=16×(2−1).
Example 3
f(x)=6cos(3x)+3x, with x in radians. Find f′(6π).
Show worked answer
Answer: −15
- f′(x)=−18sin(3x)+3.
- At x=6π,sin(3x)=sin(2π)=1.
- So f′(6π)=−18×(1)+3=−15.