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🔴Explanations to Question 3, 4 and 5👆👆 ========================= #Calculus #ESSLCE #ExitExam ========================= Question ️3️⃣ To find the derivative of the function f(x) = 3x^4 - 2x^3 + 5x - 1, we need to use the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1). ✅Using the power rule, we can find the derivative of each term in the function f(x) and add them up to get the derivative of the whole function. So, we have: f'(x) = d/dx (3x^4) - d/dx (2x^3) + d/dx (5x) - d/dx (1) Using the power rule, we get: f'(x) = 12x^3 - 6x^2 + 5 - 0 Simplifying, we get: f'(x) = 12x^3 - 6x^2 + 5 📝Therefore, the derivative of the function f(x) = 3x^4 - 2x^3 + 5x - 1 is f'(x) = 12x^3 - 6x^2 + 5. Question ️4️⃣ The integral of the function f(x) = 2x + 3 is: ∫ (2x + 3) dx ✅️The power rule states that if f(x) = x^n, then ∫ f(x) dx = x^(n+1)/(n+1) + C, where C is the constant of integration. So, integrating 2x with respect to x, we get: ∫ 2x dx = x^2 + C1 Integrating 3 with respect to x, we get: ∫ 3 dx = 3x + C2 Putting these together, we get: ∫ (2x + 3) dx = x^2 + 3x + C where C = C1 + C2 is the constant of integration. Therefore, the integral of the function f(x) = 2x + 3 is x^2 + 3x + C. Question5️⃣ To find the derivative of the natural logarithmic function f(x) = ln(5x + 2), we need to use the chain rule of differentiation, which states that if f(x) = g(h(x)), then f'(x) = g'(h(x))*h'(x). In this case, g(x) = ln(x) and h(x) = 5x + 2. So, we have: f(x) = g(h(x)) = ln(5x + 2) Using the chain rule, we get: f'(x) = g'(h(x)) h'(x) To find g'(x), we can use the fact that d/dx (ln(x)) = 1/x. So, we have: g'(x) = d/dx (ln(x)) = 1/x To find h'(x), we can use the power rule of differentiation. So, we have: h'(x) = d/dx (5x + 2) = 5 Putting it all together, we get: f'(x) = g'(h(x))  h'(x) = (1/(5x + 2)) 5 Simplifying, we get: f'(x) = 5/(5x + 2) Therefore, the derivative of the natural logarithmic function f(x) = ln(5x + 2) is f'(x) = 5/(5x + 2)🟢 ≈============== (EXIT EXAM PREPARATION HELP) ≈============== SHARE 👇👇👇 https://t.me/tutorialpointeth

🔴Explanation to Question 2️⃣ 👆For a polynomial function, we can find the limit of the function f(x) as x approaches 2 by direct substitution, but (for rational functions)⁉️ direct substitution will result in an undefined expression because the denominator of the function becomes zero at x = 2. Therefore, we need to use algebraic manipulation to simplify the expression and evaluate the limit. 🟢We can factor the numerator of the function using the product-sum formula: x^2 - 3x + 2 = (x - 2)(x - 1) Then, we can simplify the expression by canceling out the common factor of (x - 2) in the numerator and denominator: f(x) = (x^2 - 3x + 2)/(x - 2) = (x - 2)(x - 1)/(x - 2) = x - 1 Now, we can evaluate the limit of the simplified expression as x approaches 2 by direct substitution: lim x->2 (x - 1) = 2 - 1 = 1🔴 Therefore, the limit of the function f(x) as x approaches 2 is 1.