find d/dx (integral from 2 to x^4) tan(x^2) dx

tan(x^4)^2*4x^3 MY ANSWER

I assume you meant

d/dx ∫[2,x^4] tan(t^2) dt
= tan((x^4)^2) * 4x^3
= 4x^3 tan(x^8)

yes! thank you!

To find the derivative of the integral ∫(2 to x^4) tan(x^2) dx with respect to x, we can use the Fundamental Theorem of Calculus.

According to the theorem, if we have a function f(t) that is continuous on the interval [a, b], then the derivative of the integral ∫(a to x) f(t) dt with respect to x is given by f(x).

In this case, let's denote the integral as I(x):
I(x) = ∫(2 to x^4) tan(x^2) dx

Now, we can find the derivative of I(x) using the chain rule. Let's denote x^4 as u and tan(x^2) as v:
I(x) = ∫(2 to u) v du

Applying the Fundamental Theorem of Calculus, we have:
d/dx [I(x)] = v(x^4) * d/dx [x^4]
= tan(x^2) * 4x^3

Therefore, the derivative of the integral ∫(2 to x^4) tan(x^2) dx with respect to x is 4x^3 * tan(x^2).

To find the derivative of the integral, you can use the Fundamental Theorem of Calculus. Let's break down the steps to find the derivative of the integral ∫(2 to x^4) tan(x^2) dx.

Step 1: Apply the Fundamental Theorem of Calculus.
According to the theorem, the derivative of the integral of a function f(x) with respect to x is simply the integrand evaluated at the upper limit of the integral, multiplied by the derivative of the upper limit. Mathematically, it can be represented as:
d/dx ∫(a to g(x)) f(t) dt = f(g(x)) * g'(x)

Step 2: Apply the chain rule.
Since the upper limit of the integral is x^4, we need to apply the chain rule to differentiate it. The chain rule states that the derivative of a composition of functions is equal to the derivative of the outer function multiplied by the derivative of the inner function.

Let y = x^4, then dy/dx = 4x^3 (by applying the power rule).

Step 3: Apply the Fundamental Theorem of Calculus.
Substituting the evaluated integrand and the derivative of the upper limit into the Fundamental Theorem of Calculus formula, we have:
d/dx ∫(2 to x^4) tan(x^2) dx = tan(x^2) * d/dx (x^4)
= tan(x^2) * 4x^3

Therefore, the correct answer is d/dx ∫(2 to x^4) tan(x^2) dx = tan(x^2) * 4x^3.