The image contains three mathematical expressions for which we need to find the gradient (or derivative). Let's analyze and solve each one.

1. $y = \sin(x) + \cos(x)$
2. $y = e^{2x} + x$
3. $y = 2 \ln(x) + x^2$

Let's find the gradient for each function.

1. $y = \sin(x) + \cos(x)$

The gradient (or derivative) of $y$ with respect to $x$ is: $\frac{dy}{dx} = \frac{d}{dx} (\sin(x) + \cos(x))$

Using the derivatives of sine and cosine: $\frac{dy}{dx} = \cos(x) - \sin(x)$

2. $y = e^{2x} + x$

The gradient (or derivative) of $y$ with respect to $x$ is: $\frac{dy}{dx} = \frac{d}{dx} (e^{2x} + x)$

Using the chain rule for the derivative of $e^{2x}$: $\frac{dy}{dx} = 2e^{2x} + 1$

3. $y = 2 \ln(x) + x^2$

The gradient (or derivative) of $y$ with respect to $x$ is: $\frac{dy}{dx} = \frac{d}{dx} (2 \ln(x) + x^2)$

Using the derivatives of natural logarithm and power functions: $\frac{dy}{dx} = \frac{2}{x} + 2x$

Summary

1. For $y = \sin(x) + \cos(x)$: $\frac{dy}{dx} = \cos(x) - \sin(x)$

2. For $y = e^{2x} + x$: $\frac{dy}{dx} = 2e^{2x} + 1$

3. For $y = 2 \ln(x) + x^2$: $\frac{dy}{dx} = \frac{2}{x} + 2x$

Would you like further details or have any questions?

Related Questions

1. How do you find the second derivative of $y = \sin(x) + \cos(x)$?
2. What is the gradient of $y = \tan(x)$?
3. How do you find the gradient of $y = \ln(x) + e^x$?
4. What is the gradient of $y = x^3 + 3x^2 + 5$?
5. How do you determine the gradient of $y = e^{3x} + \ln(x)$?
6. What is the derivative of $y = \arcsin(x) + \arccos(x)$?
7. How do you compute the gradient of $y = \sqrt{x} + \frac{1}{x}$?
8. What is the gradient of $y = x \ln(x)$?

Tip

When differentiating composite functions, remember to use the chain rule, which states that $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$.