# How do you differentiate #G(x) = intsqrtt sint dt# from #sqrt(x)# to #x^3#?

You have

By signing up, you agree to our Terms of Service and Privacy Policy

To differentiate ( G(x) = \int_{\sqrt{t}}^{\sin(t)} dt ) from ( \sqrt{x} ) to ( x^3 ), we first need to differentiate the integral expression with respect to ( x ) using the Fundamental Theorem of Calculus. Then, we differentiate ( \sqrt{x} ) and ( x^3 ) separately with respect to ( x ). Let's denote the integral function as ( F(x) ).

The Fundamental Theorem of Calculus states that if ( F(x) = \int_{a(x)}^{b(x)} f(t) dt ), then ( F'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) ).

Differentiating ( F(x) ), we get:

[ F'(x) = \sin(\sin(x)) \cdot \frac{d}{dx}(\sin(x)) - \sin(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) ]

For ( \sqrt{x} ), the derivative is ( \frac{1}{2\sqrt{x}} ).

For ( x^3 ), the derivative is ( 3x^2 ).

Therefore, we have:

[ F'(x) = \sin(\sin(x)) \cdot \cos(x) - \frac{\sin(\sqrt{x})}{2\sqrt{x}} ]

To differentiate ( \sqrt{x} ), we simply apply the power rule, which gives us ( \frac{1}{2\sqrt{x}} ).

To differentiate ( x^3 ), we apply the power rule again, which gives us ( 3x^2 ).

Thus, we differentiate ( G(x) = \int_{\sqrt{t}}^{\sin(t)} dt ) with respect to ( x ) to get ( \sin(\sin(x)) \cdot \cos(x) - \frac{\sin(\sqrt{x})}{2\sqrt{x}} ).

The differentiation of ( \sqrt{x} ) with respect to ( x ) is ( \frac{1}{2\sqrt{x}} ), and the differentiation of ( x^3 ) with respect to ( x ) is ( 3x^2 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7