Question

Y=sin2xcosx; y=(x+2)^5(x-3)^4 (ПОМОГИТЕЕ ПЛИИИЗ) НАЙДИТЕ ПРОИЗВОДНУЮ ФУНКЦИИ

Answer 1

1)d/dx(sin2x*cosx)=cos2x*2cosx+sin2x*(-sinx)=cos2x*2cosx-sin2x*sinx)
2)d/dx((x+2)^5(x-3)^4)=5(x+2)^4(x-3)^4+(4x^3-36x^2+108x-108)(x+2)^5

Answer 2

Finding the Derivative of the Function

To find the derivative of the function y = (x+2)^5(x-3)^4, we can use the product rule and the chain rule. Let's break it down step by step.

1. Apply the product rule: - The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by: (u(x)v(x))' = u'(x)v(x) + u(x)v'(x).

2. Identify the two functions: - In our case, u(x) = (x+2)^5 and v(x) = (x-3)^4.

3. Find the derivatives of u(x) and v(x): - To find the derivative of u(x), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative is given by: (f(g(x)))' = f'(g(x)) * g'(x).

- Applying the chain rule to u(x) = (x+2)^5, we have: - Let f(u) = u^5 and g(x) = x+2. - The derivative of f(u) = u^5 with respect to u is f'(u) = 5u^4. - The derivative of g(x) = x+2 with respect to x is g'(x) = 1. - Therefore, the derivative of u(x) = (x+2)^5 with respect to x is: u'(x) = f'(g(x)) * g'(x) = 5(x+2)^4 * 1 = 5(x+2)^4.

- Similarly, to find the derivative of v(x) = (x-3)^4, we have: - Let f(u) = u^4 and g(x) = x-3. - The derivative of f(u) = u^4 with respect to u is f'(u) = 4u^3. - The derivative of g(x) = x-3 with respect to x is g'(x) = 1. - Therefore, the derivative of v(x) = (x-3)^4 with respect to x is: v'(x) = f'(g(x)) * g'(x) = 4(x-3)^3 * 1 = 4(x-3)^3.

4. Apply the product rule: - Now that we have the derivatives of u(x) and v(x), we can apply the product rule to find the derivative of y = (x+2)^5(x-3)^4. - Using the product rule, we have: y' = u'(x)v(x) + u(x)v'(x) = 5(x+2)^4(x-3)^4 + (x+2)^5 * 4(x-3)^3.

5. Simplify the expression: - To simplify the expression further, we can factor out common terms: y' = (x+2)^4(x-3)^3 [5(x-3) + 4(x+2)].

6. Final result: - Therefore, the derivative of the function y = (x+2)^5(x-3)^4 is: y' = (x+2)^4(x-3)^3 [5(x-3) + 4(x+2)].

I hope this helps! Let me know if you have any further questions.