Solve any maths problem

Equations, derivatives, integrals, matrices, triangles, primes, statistics, or a word problem the tutor breaks into parts.

Calculus · Derivatives

Differentiate sin(3x)·cos(3t)

\frac{d^2}{dt^2}\left[\sin{\left(3 x \right)} \cos{\left(3 t \right)}\right]

Step by step

  1. \frac{d}{dt}\left[\sin{\left(3 x \right)} \cos{\left(3 t \right)}\right]

    Differentiate 2 times, one derivative at a time.

  2. \sin{\left(3 x \right)} \frac{d}{d t} \cos{\left(3 t \right)}

    Constant multiple rule: pull the constant out.

  3. - \sin{\left(3 t \right)} \sin{\left(3 x \right)} \frac{d}{d t} 3 t

    Chain rule: (cos u)′ = −sin u, times u′ where u = 3 t.

  4. - 3 \sin{\left(3 t \right)} \sin{\left(3 x \right)} \frac{d}{d t} t

    Constant multiple rule: pull the constant out.

  5. - 3 \sin{\left(3 t \right)} \sin{\left(3 x \right)}

    d/dx of x is 1.

  6. f^{(1)}(t) = - 3 \sin{\left(3 t \right)} \sin{\left(3 x \right)}

    That is derivative number 1.

  7. - 3 \sin{\left(3 x \right)} \frac{d}{d t} \sin{\left(3 t \right)}

    Constant multiple rule: pull the constant out.

  8. - 3 \sin{\left(3 x \right)} \cos{\left(3 t \right)} \frac{d}{d t} 3 t

    Chain rule: (sin u)′ = cos u, times u′ where u = 3 t.

  9. - 9 \sin{\left(3 x \right)} \cos{\left(3 t \right)} \frac{d}{d t} t

    Constant multiple rule: pull the constant out.

  10. - 9 \sin{\left(3 x \right)} \cos{\left(3 t \right)}

    d/dx of x is 1.

  11. f^{(2)}(t) = - 9 \sin{\left(3 x \right)} \cos{\left(3 t \right)}

    That is derivative number 2.

Reveal the answer
f''(t) = - 9 \sin{\left(3 x \right)} \cos{\left(3 t \right)}