Question

Derivatives of integrals. Simplify the following expression. See attachment.

Answer 1

chain rule for this one. the derivative of the integral is the integrand, so you get
\[\sin(x^3)\times 3x^2\]

Answer 2

Hmm, so you don't find the derivative of sin(t)?
sin(x3)×3^x2, is the answer?

Answer 3

oh no, you do not find the derivative of
\[\sin(t)\]
\[F(x)=\int_a^xf(t)dt\] then
\[F'(x)=f(x)\]

Answer 4

in this case you have a composite function
\[F(x^3)=\int_a^{x^3}\sin(t)dt\] so
\[F'(x^3)=\sin(x^3)\times 3x^2\] via the chain rule

Answer 5

Makes perfect sense now, thanks!

Answer 6

yw