Differentiate ( 4x+2)_3 with respect to x .hence,find the value of dy/ DX when x=1
I'll assume you meant
y = (4x+2)^3
If so, then by the chain rule,
dy/dx = 3(4x+2)^2 * 4
so, y'(1) = 3*6^2*4 = 288
Well, if we apply the power rule, the differentiation of (4x + 2)^3 would be 3(4x + 2)^2 multiplied by the derivative of 4x + 2, which is just 4. So, the result is 3(4x + 2)^2 * 4.
Now, to find dy/dx when x = 1, we substitute x = 1 into our derivative expression. Then we get 3(4(1) + 2)^2 * 4, which simplifies to 3(6)^2 * 4. After some calculations, we find that dy/dx when x = 1 is equal to... 432. And that's the value we're looking for!
Remember, though, I hope you enjoyed this crazy rollercoaster ride of derivatives and numbers. Feel free to ask anything else!
To differentiate (4x + 2)^3 with respect to x, we will use the chain rule. The chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
Let's break it down:
Step 1: Find dy/du
We will first differentiate (4x + 2)^3 with respect to u.
Let u = 4x + 2
Therefore, (4x + 2)^3 can be rewritten as u^3.
Now, differentiating u^3 with respect to u gives us:
du^3/du = 3u^2
So, dy/du = 3u^2
Step 2: Find du/dx
To find du/dx, we need to differentiate u = 4x + 2 with respect to x.
Differentiating 4x + 2 with respect to x gives us:
du/dx = 4
Step 3: Find dy/dx
Using the chain rule, we multiply dy/du by du/dx.
dy/dx = (dy/du) * (du/dx)
dy/dx = (3u^2) * (4)
Step 4: Evaluate dy/dx at x = 1
Since we need to find dy/dx when x = 1, we substitute u = 4x + 2 = 4(1) + 2 = 6.
Therefore,
dy/dx = (3u^2) * (4)
dy/dx = 3(6^2) * 4
dy/dx = 3(36) * 4
dy/dx = 432
Hence, when x = 1, the value of dy/dx is 432.
To differentiate the expression (4x + 2)³ with respect to x, we can use the power rule of differentiation. The power rule states that for any term of the form (ax + b)ⁿ, the derivative is equal to n*(ax + b)^(n-1) * a.
Let's apply the power rule to differentiate (4x + 2)³:
Step 1: Identify a = 4, b = 2, and n = 3.
Step 2: Apply the power rule: (4x + 2)³ = 3 * (4x + 2)² * 4.
Simplifying further:
= 12 * (4x + 2)²
Now we have obtained the derivative of (4x + 2)³ with respect to x, which is 12*(4x + 2)².
To find the value of dy/dx when x = 1, substitute x = 1 into the derivative we found:
dy/dx = 12*(4(1) + 2)²
Simplifying further:
= 12*(6)²
= 12 * 36
= 432
So, the value of dy/dx when x = 1 is 432.