3 Differentiate the following with respect to x y = π ln x + 2.3⁵ˣ + 4/cotx + 2log₃x + 1/4x + 1/x⁶

2.2 Expand to THREE terms only with the aid of the binomial theorem. Step 1: Rewrite the expression in the standard binomial form (a+b)^n. The given expression is (1)/((4x+6)^-6). This can be rewritten as (4x+6)^6. Here, a = 4x, b = 6, and n = 6. The binomial theorem states that (a+b)^n = n0a^n b^0 + n1a^n-1b^1 + n2a^n-2b^2 + Step 2: Calculate the first term (k=0). 60(4x)^6(6)^0 = 1 · (4^6 x^6) · 1 = 4096x^6 Step 3: Calculate the second term (k=1). 61(4x)^5(6)^1 = 6 · (4^5 x^5) · 6 = 36 · (1024 x^5) = 36864x^5 Step 4: Calculate the third term (k=2). 62(4x)^4(6)^2 = (6 × 5)/(2 × 1) · (4^4 x^4) · 36 = 15 · (256 x^4) · 36 = 138240x^4 The expansion to three terms is: 4096x^6 + 36864x^5 + 138240x^4 2.3 Differentiate the following with respect to x. y = x + 2 · 3^5x + (4)/( x) + 2 _3 x + (1)/(4x) + (1)/(x^6) Step 1: Rewrite terms for easier differentiation. y = x + 2 · 3^5x + 4 x + (2 x)/( 3) + (1)/(4)x^-1 + x^-6 Step 2: Differentiate each term using the appropriate rules. • For x: (d)/(dx)( x) = · (1)/(x) = ()/(x) • For 2 · 3^5x: Use the rule (d)/(dx)(a^kx) = k · a^kx a. (d)/(dx)(2 · 3^5x) = 2 · (5 · 3^5x 3) = 10 · 3^5x 3 • For 4 x: (d)/(dx)(4 x) = 4 ^2 x • For (2 x)/( 3): (d)/(dx)((2)/( 3) x) = (2)/( 3) · (1)/(x) = (2)/(x 3) • For (1)/(4)x^-1: (d)/(dx)((1)/(4)x^-1) = (1)/(4)(-1)x^-2 = -(1)/(4x^2) • For x^-6: (d)/(dx)(x^-6) = -6x^-7 = -(6)/(x^7) Step 3: Combine all the differentiated terms. (dy)/(dx) = ()/(x) + 10 · 3^5x 3 + 4 ^2 x + (2)/(x 3) - (1)/(4x^2) - (6)/(x^7) The final answer is: (dy)/(dx) = ()/(x) + 10 · 3^5x 3 + 4 ^2 x + (2)/(x 3) - (1)/(4x^2) - (6)/(x^7) 2.4 Differentiate with the aid of the quotient rule. y = (4 x)/(10^2x) Step 1: Identify u and v for the quotient rule (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Let u = 4 x and v = 10^2x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(4 x) = -4 x v' = (d)/(dx)(10^2x). Using the rule (d)/(dx)(a^kx) = k · a^kx a, we get: v' = 2 · 10^2x 10 Step 3: Apply the quotient rule formula. (dy)/(dx) = (-4 x)(10^2x) - (4 x)(2 · 10^2x 10)(10^2x)^2 Step 4: Simplify the expression. (dy)/(dx) = -4 · 10^2x x - 8 · 10^2x x 1010^4x Factor out 4 · 10^2x from the numerator: (dy)/(dx) = 4 · 10^2x (- x - 2 x 10)10^4x Cancel 10^2x from the numerator and denominator: (dy)/(dx) = (4 (- x - 2 x 10))/(10^2x) (dy)/(dx) = -(4 ( x + 2 x 10))/(10^2x) The final answer is: (dy)/(dx) = -(4 ( x + 2 x 10))/(10^2x) What's next? 📸