PIL import Image import pytesseract

Load the uploaded image

image_path = "/mnt/data/1733670422069e5vz2j0b.jpg" img = Image.open(image_path)

Perform OCR to extract text

extracted_text = pytesseract.image_to_string(img, lang='eng')

extracted_textThe OCR output from the image was slightly inaccurate due to the quality of the text. After manually reviewing and correcting it, here is the extracted and corrected content:

Functions to Differentiate:

1. $f(x) = 3x^8 - 7x - 2.5$
2. $f(x) = \frac{6}{x^4} - \frac{x^5}{2} + 2\sqrt{x}$
3. $f(x) = \cos x (5 - x^2)$
4. $f(x) = \frac{x + 2}{\sin x}$
5. $f(x) = (5x^2 - 1) \cdot (x + 4)$
6. $f(x) = (3x^5 - 2)^8$
7. $f(x) = \cos(5x)$

I will now compute the derivatives for each function. sympy import symbols, diff, cos, sin, sqrt

Define the variable

x = symbols('x')

Functions

f1 = 3x**8 - 7x - 2.5 f2 = 6/x4 - x5/2 + 2sqrt(x) f3 = cos(x) * (5 - x**2) f4 = (x + 2) / sin(x) f5 = (5x2 - 1) * (x + 4) f6 = (3*x5 - 2)**8 f7 = cos(5*x)

Derivatives

derivatives = { "f1'": diff(f1, x), "f2'": diff(f2, x), "f3'": diff(f3, x), "f4'": diff(f4, x), "f5'": diff(f5, x), "f6'": diff(f6, x), "f7'": diff(f7, x) }

derivativesHere are the derivatives of the given functions:

1. $f'(x) = 24x^7 - 7$
2. $f'(x) = -\frac{24}{x^5} - \frac{5x^4}{2} + \frac{1}{\sqrt{x}}$
3. $f'(x) = -2x\cos(x) - (5 - x^2)\sin(x)$
4. $f'(x) = \frac{1}{\sin(x)} - \frac{(x + 2)\cos(x)}{\sin^2(x)}$
5. $f'(x) = 10x(x + 4) + 5x^2 - 1$
6. $f'(x) = 120x^4(3x^5 - 2)^7$
7. $f'(x) = -5\sin(5x)$

Would you like a detailed explanation of any particular derivative or step? Let me know!

Questions to Explore Further:

1. What rules are applied to calculate the derivative of $f_6(x)$?
2. How does the product rule simplify $f_5(x)$?
3. Can you compute the second derivative for any of the functions?
4. How do trigonometric functions' derivatives behave in $f_7(x)$?
5. Why does $f_4(x)$ involve the quotient rule?

Tip:

Remember, always check your function for simplifications before differentiation to make the process easier!