Question

Find
`(dy)/(dx)` if
`y=\int\limits^a_b {e^(t)tan } \,t dt`, where a=x^3, b=3x

Answer 1

3[x²(e^x³)(tan(x³) - (e^3x)tan(3x)]

Explanation:

Derivative of x³ = 3x²

Derivative of 3x = 3

3x²(e^x³)(tan(x³) - 3(e^3x)tan(3x)

3[x²(e^x³)(tan(x³) - (e^3x)tan(3x)]

Answer 2

Explanation:

Use the second fundamental theorem of calculus.

If y = ∫ₐᵇ f(t) dt, then dy/dx = f(b) db/dx − f(a) da/dx

dy/dx = e^(x³) tan (x³) (3x²) − e^(3x) tan (3x) (3)

dy/dx = 3x² e^(x³) tan (x³) − 3 e^(3x) tan (3x)