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(a) Using the definition of the derivative, find the derivative of the function f(x) = 3x² + 1 f (x) = h / f (x-h ) - f (x)

(b) Now using (a) find the equation of the tangent line to the curve y = f(x) at (1,2).

User Mati
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Final answer:

To find the derivative of the function f(x) = 3x² + 1, use the definition of the derivative by taking the limit as h approaches 0. The derivative is 6x. To find the equation of the tangent line to the curve y = f(x) at (1, 2), use the point-slope form of a linear equation. The equation of the tangent line is y = 6x - 4.

Step-by-step explanation:

To find the derivative of the function f(x) = 3x² + 1 using the definition of the derivative, we can use the following formula:

f'(x) = lim(h→0) [f(x + h) - f(x)] / h

Substitute the given function f(x) with 3x² + 1:

f'(x) = lim(h→0) [(3(x + h)² + 1) - (3x² + 1)] / h

Expand and simplify the expression:

f'(x) = lim(h→0) [(3x² + 6hx + 3h² + 1) - (3x² + 1)] / h

Combine like terms:

f'(x) = lim(h→0) [6hx + 3h²] / h

Cancel out the h in the numerator and denominator:

f'(x) = lim(h→0) [6x + 3h]

Take the limit as h approaches 0:

f'(x) = 6x

Therefore, the derivative of f(x) = 3x² + 1 is f'(x) = 6x.

To find the equation of the tangent line to the curve y = f(x) at (1, 2), we will use the point-slope form of a linear equation. We know that the slope of the tangent line is equal to the derivative of f(x) at x = 1, which is 6. Now we have the slope (m = 6) and a point (1, 2) on the line, so we can substitute these values into the point-slope form equation:

y - y₁ = m(x - x₁)

Substitute y₁ = 2, x₁ = 1, and m = 6:

y - 2 = 6(x - 1)

Simplify the equation:

y - 2 = 6x - 6

Add 2 to both sides of the equation:

y = 6x - 4

Therefore, the equation of the tangent line to the curve y = f(x) at (1, 2) is y = 6x - 4.

User Pa Ye
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