https://file.aiquickdraw.com/mathbot/file/1722776863196e1kxz7p1.jpg

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 \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).

Learn how to find the gradient (derivative) of sine, cosine, exponential, and logarithmic functions with detailed examples and solutions. This page covers differentiation rules for trigonometric, exponential, and logarithmic functions, including the chain rule. Suitable for college-level students and anyone interested in calculus.

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