How To Find The Derivative Of A Function And Use It To Find Slope And Tangent Line Equation

Find derivative of f(x), plug in point to find slope. Plug into y – y1 = m(x – x1).

Find the equation of a line tangent to f(x).

To find the derivative of a function f(x), we need to take the derivative of f(x) with respect to x. This can be denoted as f'(x) or dy/dx. The derivative of f(x) gives us the slope of the function at any given point.

For example, let’s say we have the function f(x) = 2x^2 + 3x + 4. To find the derivative of f(x), we need to take the derivative of each term using the power rule:

f(x) = 2x^2 + 3x + 4
f'(x) = 4x + 3

Now, let’s say we want to find the slope of the function at the point x=2. To do this, we plug in x=2 into the derivative we just found:

f'(2) = 4(2) + 3
f'(2) = 11

So the slope of the function at x=2 is 11.

To find the equation of the tangent line at x=2, we use the point-slope form of a line:

y – y1 = m(x – x1)

We know x1=2 and y1=f(2), so we can plug these values in:

y – f(2) = 11(x-2)

Now we just need to simplify and solve for y:

y – (2(2)^2 + 3(2) + 4) = 11x – 22
y – 16 = 11x – 22
y = 11x – 6

So the equation of the tangent line at x=2 is y=11x-6.

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