Compute the derivative of: ƒ(x) = (x^3 + 2x)e^x

Question

e is Euler's number.

Compute the derivative of:

ƒ(x) = (x³ + 2x)e^x

Compute the integral:

∫ e^x(4 + e^x)⁵ dx

Compute the limit:

lim x→+∞ ((e^4x - e^-4x)/(e^4x + e^-4x))

Answer 1

1) Compute the derivative of f(x) = (x³+2x)e^x

Since we have two functions being multiplied, we can use the Product Rule, which states that d/dx f(x)g(x) = f'(x)g(x) + f(x)g'(x). Let f(x) = (x³+2x), f'(x) = 3x²+2, g(x) = g'(x) = e^x:

d/dx (x³+2x)e^x = (3x²+2)e^x + (x³+2x)e^x = (3x²+2+x³+2x)e^x = (x³+3x²+2x+2)e^x

2) Compute the integral ∫e^x(4+e^x)⁵ dx

We can use u-substitution here to reduce the number of terms while doing integration. Let u=4+e^x and du=e^xdx:

∫e^x(4+e^x)⁵ dx = ∫u⁵ du = u⁶/6 + C = (4+e^x)⁶/6 + C

3) Compute the limit lim x→+∞ ((e^4x - e^-4x)/(e^4x + e^-4x))

In this problem, we can rewrite the function as follows:

lim x→+∞ ((e^4x - e^-4x)/(e^4x + e^-4x))

lim x→+∞ [e^4x(1-e^-8x)]/[e^4x(1+e^-8x)]

lim x→+∞ (1-e^-8x)/(1+e^-8x)

Now it's easy to see as x goes to infinity, e^-8x in both the numerator and the denominator go to 0, so the limit becomes:

lim x→+∞ (1-e^-8x)/(1+e^-8x)

= (1-0)/(1+0)

= 1/1

= 1

I hope these answers and explanations helped!