Let’s solve the problems from the image step-by-step.

1. Find the derivative (dy/dx) of the following functions:

(a) $y = x^5$

• Using the power rule ($\frac{d}{dx}[x^n] = nx^{n-1}$): $\frac{dy}{dx} = 5x^{4}$

(b) $y = e^{4x}$

• Using the chain rule for $e^{u}$, where $u = 4x$: $\frac{dy}{dx} = 4e^{4x}$

(c) $y = \ln x$

• The derivative of $\ln x$ is: $\frac{dy}{dx} = \frac{1}{x}$

(d) $y = \sin(2x)$

• Using the chain rule for $\sin(u)$, where $u = 2x$: $\frac{dy}{dx} = 2\cos(2x)$

2. Evaluate the following integrals:

(a) $\int 15x \, dx$

• Using the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ $\int 15x \, dx = 15 \cdot \frac{x^{2}}{2} + C = \frac{15x^2}{2} + C$

(b) $\int (3x^2 - 4) \, dx$

• Split the integral and apply the power rule: $\int 3x^2 \, dx - \int 4 \, dx = 3 \cdot \frac{x^3}{3} - 4x + C = x^3 - 4x + C$

(c) $\int \frac{3}{x^4} \, dx$

• Rewrite $\frac{1}{x^4}$ as $x^{-4}$ and apply the power rule: $\int 3x^{-4} \, dx = 3 \cdot \frac{x^{-3}}{-3} + C = -x^{-3} + C = -\frac{1}{x^3} + C$

(d) $\int (5x^2 + 2\cos(x)) \, dx$

• Split the integral: $\int 5x^2 \, dx + \int 2\cos(x) \, dx$
  • For $\int 5x^2 \, dx$: $5 \cdot \frac{x^3}{3} = \frac{5x^3}{3}$
  • For $\int 2\cos(x) \, dx$: $2 \sin(x)$
  • Combining: $\frac{5x^3}{3} + 2\sin(x) + C$

Final Results:

1. Derivatives:

   • (a) $\frac{dy}{dx} = 5x^4$
   • (b) $\frac{dy}{dx} = 4e^{4x}$
   • (c) $\frac{dy}{dx} = \frac{1}{x}$
   • (d) $\frac{dy}{dx} = 2\cos(2x)$

2. Integrals:

   • (a) $\int 15x \, dx = \frac{15x^2}{2} + C$
   • (b) $\int (3x^2 - 4) \, dx = x^3 - 4x + C$
   • (c) $\int \frac{3}{x^4} \, dx = -\frac{1}{x^3} + C$
   • (d) $\int (5x^2 + 2\cos(x)) \, dx = \frac{5x^3}{3} + 2\sin(x) + C$

Do you want a deeper explanation or any clarifications?

Expanding Questions:

1. How is the power rule applied to differentiation and integration?
2. What is the chain rule, and when do we use it in differentiation?
3. Why do we add a constant $C$ after integration?
4. What are the specific conditions for using logarithmic differentiation?
5. How can trigonometric integrals like $\int \cos(x) \, dx$ be solved?

Tip:

Always check if the function has a constant multiplier or requires a substitution before directly applying the differentiation or integration rules!