What is an equation of the line tangent to the graph of f(x)=x2(2x+1)3 at the point where x=-1?
(that's x squared (2x+1)cubed)
I'm having trouble finding the derivative:f'(x)which would be the slope (m).
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To find the equation of the line tangent to the graph of the function f(x) = x^2(2x+1)^3 at the point where x = -1, we first need to find the slope of the tangent line, which is the derivative of the function f(x) at x = -1.
To find the derivative, f'(x), of the given function, we can use the product rule and chain rule of differentiation. Let's break it down step by step:
Step 1: Find the derivative of the first term, x^2, with respect to x.
The derivative of x^2 can be found using the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1).
So, the derivative of x^2 with respect to x is 2x.
Step 2: Find the derivative of the second term, (2x+1)^3, with respect to x.
To differentiate this term, we can use the chain rule. The chain rule states that if g(x) = f(h(x)), then g'(x) = f'(h(x)) * h'(x).
Here, f(x) = x^3, which has a derivative of 3x^2, and h(x) = 2x+1.
Using the chain rule, the derivative of (2x+1)^3 is 3(2x + 1)^2 * (2), since the derivative of 2x+1 is 2.
Step 3: Apply the product rule to find the derivative of the entire function f(x).
The product rule states that if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).
In this case, g(x) = x^2 and h(x) = (2x+1)^3.
So, the derivative of f(x) = g(x) * h(x) is given by f'(x) = g'(x) * h(x) + g(x) * h'(x).
Substituting the derivatives we found in the previous steps, we have:
f'(x) = 2x * (2x+1)^3 + x^2 * 3(2x + 1)^2 * (2)
Simplifying this expression, we get the derivative f'(x).
Now, to find the slope of the tangent line at x = -1, substitute x = -1 into the derivative f'(x) that we just found. Evaluate the expression at x = -1 to get the slope of the tangent line.
Finally, we can find the equation of the tangent line using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point (-1, f(-1)) and m is the slope of the tangent line.
Plug in the values of x1, y1, and m into the equation to get the equation of the tangent line in the form y = mx + b, where b is the y-intercept.