1. Find $\frac{dy}{dx}$ if $y = 4x^7 + 3 \cos 2x - \log x$. Step 1: Differentiate each term separately. - $\frac{d}{dx}(4x^7) = 28x^6$ - $\frac{d}{dx}(3 \cos 2x) = 3 \cdot (-\sin 2x) \cdot 2 = -6 \sin 2x$ - $\frac{d}{dx}(-\log x) = -\frac{1}{x}$ Step 2: Combine results. $$\frac{dy}{dx} = 28x^6 - 6 \sin 2x - \frac{1}{x}$$ 2. Find $\frac{dy}{dx}$ if $y = 5 e^{6x} + 8x^5 - 2^x + 3 \sin 3x - 8$. Step 1: Differentiate each term. - $\frac{d}{dx}(5 e^{6x}) = 5 \cdot 6 e^{6x} = 30 e^{6x}$ - $\frac{d}{dx}(8x^5) = 40x^4$ - $\frac{d}{dx}(-2^x) = -2^x \ln 2$ - $\frac{d}{dx}(3 \sin 3x) = 3 \cdot 3 \cos 3x = 9 \cos 3x$ - $\frac{d}{dx}(-8) = 0$ Step 2: Combine results. $$\frac{dy}{dx} = 30 e^{6x} + 40x^4 - 2^x \ln 2 + 9 \cos 3x$$ 3. Find $\frac{dy}{dx}$ if $y = \frac{3}{x^2} + 4 e^x - \sec x + 5^x - 10$. Step 1: Rewrite $\frac{3}{x^2} = 3x^{-2}$. Step 2: Differentiate each term. - $\frac{d}{dx}(3x^{-2}) = 3 \cdot (-2) x^{-3} = -6x^{-3}$ - $\frac{d}{dx}(4 e^x) = 4 e^x$ - $\frac{d}{dx}(-\sec x) = -\sec x \tan x$ - $\frac{d}{dx}(5^x) = 5^x \ln 5$ - $\frac{d}{dx}(-10) = 0$ Step 3: Combine results. $$\frac{dy}{dx} = -6x^{-3} + 4 e^x - \sec x \tan x + 5^x \ln 5$$ 4. Find $\frac{dy}{dx}$ if $y = x \log x$. Step 1: Use product rule: $\frac{d}{dx}(uv) = u'v + uv'$ with $u = x$, $v = \log x$. - $u' = 1$ - $v' = \frac{1}{x}$ Step 2: Compute derivative. $$\frac{dy}{dx} = 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1$$ 5. Find $\frac{dy}{dx}$ if $y = \sin x \log x$. Step 1: Use product rule with $u = \sin x$, $v = \log x$. - $u' = \cos x$ - $v' = \frac{1}{x}$ Step 2: Compute derivative. $$\frac{dy}{dx} = \cos x \log x + \sin x \cdot \frac{1}{x} = \cos x \log x + \frac{\sin x}{x}$$ 6. Find $\frac{dy}{dx}$ if $y = (x^2 + 5)(x^3 + 3)$. Step 1: Use product rule with $u = x^2 + 5$, $v = x^3 + 3$. - $u' = 2x$ - $v' = 3x^2$ Step 2: Compute derivative. $$\frac{dy}{dx} = 2x (x^3 + 3) + (x^2 + 5) 3x^2 = 2x^4 + 6x + 3x^4 + 15x^2 = 5x^4 + 15x^2 + 6x$$ 7. Find $\frac{dy}{dx}$ if $y = (2x - 3) \cos x$. Step 1: Use product rule with $u = 2x - 3$, $v = \cos x$. - $u' = 2$ - $v' = -\sin x$ Step 2: Compute derivative. $$\frac{dy}{dx} = 2 \cos x + (2x - 3)(-\sin x) = 2 \cos x - (2x - 3) \sin x$$ 8. Find $\frac{dy}{dx}$ if $y = e^x \tan x$. Step 1: Use product rule with $u = e^x$, $v = \tan x$. - $u' = e^x$ - $v' = \sec^2 x$ Step 2: Compute derivative. $$\frac{dy}{dx} = e^x \tan x + e^x \sec^2 x = e^x (\tan x + \sec^2 x)$$ 9. Find $\frac{dy}{dx}$ if $y = x^2 \sin 3x$. Step 1: Use product rule with $u = x^2$, $v = \sin 3x$. - $u' = 2x$ - $v' = 3 \cos 3x$ Step 2: Compute derivative. $$\frac{dy}{dx} = 2x \sin 3x + x^2 \cdot 3 \cos 3x = 2x \sin 3x + 3x^2 \cos 3x$$ 10. Find $\frac{dy}{dx}$ if $y = (4x + 5) e^{5x}$. Step 1: Use product rule with $u = 4x + 5$, $v = e^{5x}$. - $u' = 4$ - $v' = 5 e^{5x}$ Step 2: Compute derivative. $$\frac{dy}{dx} = 4 e^{5x} + (4x + 5) 5 e^{5x} = 4 e^{5x} + 5(4x + 5) e^{5x} = (4 + 20x + 25) e^{5x} = (20x + 29) e^{5x}$$ 11. Find $\frac{dy}{dx}$ if $y = \frac{\cos x}{\sin x}$ by quotient rule. Step 1: Use quotient rule: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$ with $u = \cos x$, $v = \sin x$. - $u' = -\sin x$ - $v' = \cos x$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{(-\sin x)(\sin x) - (\cos x)(\cos x)}{\sin^2 x} = \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} = \frac{-1}{\sin^2 x} = -\csc^2 x$$ 12. Find $\frac{dy}{dx}$ if $y = \frac{1}{\cos x}$ by quotient rule. Step 1: Rewrite $y = \sec x$. Step 2: Derivative of $\sec x$ is $\sec x \tan x$. So, $$\frac{dy}{dx} = \sec x \tan x$$ 13. Find $\frac{dy}{dx}$ if $y = \frac{5x - 3}{x^2 + 4x}$. Step 1: Use quotient rule with $u = 5x - 3$, $v = x^2 + 4x$. - $u' = 5$ - $v' = 2x + 4$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{5(x^2 + 4x) - (5x - 3)(2x + 4)}{(x^2 + 4x)^2}$$ Step 3: Expand numerator. - $5(x^2 + 4x) = 5x^2 + 20x$ - $(5x - 3)(2x + 4) = 10x^2 + 20x - 6x - 12 = 10x^2 + 14x - 12$ Step 4: Subtract. $$5x^2 + 20x - (10x^2 + 14x - 12) = 5x^2 + 20x - 10x^2 - 14x + 12 = -5x^2 + 6x + 12$$ Step 5: Final derivative. $$\frac{dy}{dx} = \frac{-5x^2 + 6x + 12}{(x^2 + 4x)^2}$$ 14. Find $\frac{dy}{dx}$ if $y = \frac{\log x}{\sin x}$. Step 1: Use quotient rule with $u = \log x$, $v = \sin x$. - $u' = \frac{1}{x}$ - $v' = \cos x$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{\frac{1}{x} \sin x - \log x \cos x}{\sin^2 x}$$ 15. Find $\frac{dy}{dx}$ if $y = \frac{1}{\sin x}$. Step 1: Rewrite $y = \csc x$. Step 2: Derivative of $\csc x$ is $-\csc x \cot x$. So, $$\frac{dy}{dx} = -\csc x \cot x$$ 16. Find $\frac{dy}{dx}$ if $y = \frac{e^{3x} + 4}{x^2 - 4}$. Step 1: Use quotient rule with $u = e^{3x} + 4$, $v = x^2 - 4$. - $u' = 3 e^{3x}$ - $v' = 2x$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{3 e^{3x} (x^2 - 4) - (e^{3x} + 4) 2x}{(x^2 - 4)^2}$$ 17. Find $\frac{dy}{dx}$ if $y = \frac{e^x}{x^2}$. Step 1: Use quotient rule with $u = e^x$, $v = x^2$. - $u' = e^x$ - $v' = 2x$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{e^x x^2 - e^x 2x}{x^4} = \frac{e^x (x^2 - 2x)}{x^4} = e^x \frac{x^2 - 2x}{x^4} = e^x \frac{x - 2}{x^3}$$ 18. Find $\frac{dy}{dx}$ if $y = \frac{x^3}{x^2 + 3}$. Step 1: Use quotient rule with $u = x^3$, $v = x^2 + 3$. - $u' = 3x^2$ - $v' = 2x$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{3x^2 (x^2 + 3) - x^3 (2x)}{(x^2 + 3)^2} = \frac{3x^4 + 9x^2 - 2x^4}{(x^2 + 3)^2} = \frac{x^4 + 9x^2}{(x^2 + 3)^2}$$ 19. Find $\frac{dy}{dx}$ if $y = \frac{5x + 3}{2x - 1}$. Step 1: Use quotient rule with $u = 5x + 3$, $v = 2x - 1$. - $u' = 5$ - $v' = 2$ Step 2: Compute derivative. $$\frac{dy}{dx} = \frac{5(2x - 1) - (5x + 3) 2}{(2x - 1)^2} = \frac{10x - 5 - 10x - 6}{(2x - 1)^2} = \frac{-11}{(2x - 1)^2}$$ 20. Find $\frac{dy}{dx}$ if $y = \frac{1}{\log x}$. Step 1: Rewrite $y = (\log x)^{-1}$. Step 2: Use chain rule. $$\frac{dy}{dx} = - (\log x)^{-2} \cdot \frac{1}{x} = - \frac{1}{x (\log x)^2}$$ Final answers summarized: 1. $28x^6 - 6 \sin 2x - \frac{1}{x}$ 2. $30 e^{6x} + 40x^4 - 2^x \ln 2 + 9 \cos 3x$ 3. $-6x^{-3} + 4 e^x - \sec x \tan x + 5^x \ln 5$ 4. $\log x + 1$ 5. $\cos x \log x + \frac{\sin x}{x}$ 6. $5x^4 + 15x^2 + 6x$ 7. $2 \cos x - (2x - 3) \sin x$ 8. $e^x (\tan x + \sec^2 x)$ 9. $2x \sin 3x + 3x^2 \cos 3x$ 10. $(20x + 29) e^{5x}$ 11. $-\csc^2 x$ 12. $\sec x \tan x$ 13. $\frac{-5x^2 + 6x + 12}{(x^2 + 4x)^2}$ 14. $\frac{\frac{1}{x} \sin x - \log x \cos x}{\sin^2 x}$ 15. $-\csc x \cot x$ 16. $\frac{3 e^{3x} (x^2 - 4) - 2x (e^{3x} + 4)}{(x^2 - 4)^2}$ 17. $e^x \frac{x - 2}{x^3}$ 18. $\frac{x^4 + 9x^2}{(x^2 + 3)^2}$ 19. $\frac{-11}{(2x - 1)^2}$ 20. $- \frac{1}{x (\log x)^2}$