determine de value of dy/dx for the given value
a) y=(x^3)(3x+7)^2, x=-2
b) y= (2x+1)^5(3x+2)^4, x=-1
i know how to get the derivate but the exponents outside the brackets confuse me
From what you are saying about the exponents confusing you,
you clearly do not know how to get the derivatives.
did you get ...
y = x^3(3x+7)^2
dy/dx = x^3(2)(3x+7)(3) + 3x^2(3x+7)^2 , as your first-line derivative
using a combination of product rule and chain rule ???
This can of course be simplified, but since you just want the value
of dy/dx when x = -2, let's just sub that in
when x = -2
dy/dx = (-2)^3 (2)(-6+7)(3) + 3(4)(-6+7)^2
= -48 + 12
= -36
state what your derivative is for the 2nd equation.
then i dont know thats why im studying it :) how did you get the derivate of (3x+7)^2?
the chain rule says that if y = u^n
where u is a function of x, then
dy/dx = n u^(n-1) du/dx
If you look carefully at the calculated derivative, that is what mathhelper has done.
To determine the value of dy/dx for the given functions when x is a specific value, you can use the chain rule of differentiation. The chain rule states that if you have a function of the form f(g(x)), then the derivative of that function with respect to x is given by f'(g(x)) * g'(x), where f'(g(x)) represents the derivative of the outer function and g'(x) represents the derivative of the inner function.
Let's go through both parts of the question step by step:
a) y = (x^3)(3x + 7)^2, x = -2
1. Apply the power rule for differentiation:
- For the term x^3, the derivative is 3x^2.
- For the term (3x + 7)^2, consider it as the product of two functions: u = 3x + 7 and v = 3x + 7.
- Use the chain rule, where u = v = 3x + 7, to find the derivative of (3x + 7)^2.
- The derivative of u^2 is 2u * du/dx, where du/dx is the derivative of u with respect to x.
- The derivative of u = 3x + 7 is du/dx = 3.
- Therefore, the derivative of (3x + 7)^2 is 2(3x + 7) * 3.
2. Substitute the value x = -2 into the derivatives from step 1:
- For the term x^3, the derivative is 3x^2. Substituting x = -2 gives 3(-2)^2 = 12.
- For the term (3x + 7)^2, the derivative is 2(3x + 7) * 3. Substituting x = -2 gives 2(3(-2) + 7) * 3 = 45.
3. Multiply the derivatives obtained in step 2 by their respective terms in the function:
- For the term x^3, multiply the derivative 12 by the term itself: 12 * (-2)^3 = 12 * (-8) = -96.
- For the term (3x + 7)^2, multiply the derivative 45 by the term itself: 45 * (3(-2) + 7)^2 = 45 * 5^2 = 1125.
4. Add the results from step 3 to find the value of dy/dx:
- dy/dx = (-96) + 1125 = 1029.
Therefore, when x = -2, dy/dx = 1029 for the function y = (x^3)(3x + 7)^2.
b) y = (2x + 1)^5(3x + 2)^4, x = -1
Follow the same steps as described above to differentiate and substitute the value of x = -1 into the derivatives obtained:
1. Apply the power rule for differentiation for each term:
- The derivative of (2x + 1)^5 is 5(2x + 1)^4 * 2.
- The derivative of (3x + 2)^4 is 4(3x + 2)^3 * 3.
2. Substitute the value x = -1 into the derivatives obtained in step 1:
- For (2x + 1)^5, substitute x = -1 into 5(2x + 1)^4 * 2:
5(2(-1) + 1)^4 * 2 = 5(1)^4 * 2 = 10.
- For (3x + 2)^4, substitute x = -1 into 4(3x + 2)^3 * 3:
4(3(-1) + 2)^3 * 3 = 4(1)^3 * 3 = 12.
3. Multiply the derivatives obtained in step 2 by their respective terms in the function:
- Multiply the derivative 10 by (2x + 1)^5: 10 * (2(-1) + 1)^5 = 10 * (1)^5 = 10.
- Multiply the derivative 12 by (3x + 2)^4: 12 * (3(-1) + 2)^4 = 12 * (1)^4 = 12.
4. Add the results from step 3 to find the value of dy/dx:
- dy/dx = 10 + 12 = 22.
Therefore, when x = -1, dy/dx = 22 for the function y = (2x + 1)^5(3x + 2)^4.