# How do you find the equation of the line tangent to #f(x) = x^3# at x = 2?

The gradient (slope) of the tangent at any point is represented by the derivative at that point.

Hence the equation of the required tangent line to the curve is

By signing up, you agree to our Terms of Service and Privacy Policy

To find the equation of the line tangent to f(x) = x^3 at x = 2, we need to find the slope of the tangent line and the point of tangency.

First, we find the derivative of f(x) = x^3 using the power rule, which states that the derivative of x^n is n*x^(n-1).

Taking the derivative of f(x) = x^3, we get f'(x) = 3x^2.

Next, we substitute x = 2 into the derivative to find the slope of the tangent line at x = 2.

Substituting x = 2 into f'(x) = 3x^2, we get f'(2) = 3(2)^2 = 12.

So, the slope of the tangent line at x = 2 is 12.

To find the point of tangency, we substitute x = 2 into f(x) = x^3.

Substituting x = 2 into f(x) = x^3, we get f(2) = (2)^3 = 8.

Therefore, the point of tangency is (2, 8).

Using the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept, we can write the equation of the tangent line.

Substituting the slope (m = 12) and the point of tangency (x = 2, y = 8) into the slope-intercept form, we get y = 12x - 16.

Thus, the equation of the line tangent to f(x) = x^3 at x = 2 is y = 12x - 16.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you find the equations for the normal line to #x^2+y^2=9# through (0,3)?
- If the tangent line to #y = f(x)# at #(4,3)# passes through the point #(0,2)#, Find #f(4)# and #f'(4)#? An explanation would also be very helpful.
- The graph of the function (x^2 + y^2)^2 = 4x^2y is a double folium as shown below. (a) Find, algebraically, all points on the curve with y = 1? (b) Verify that the slopes of tangent lines to both points with y = 1 is equal to 0?
- How do you find f'(x) using the limit definition given # f(x)=3x^(−2)#?
- What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(2t-pi/2),cost/t ) # at # t=(3pi)/8 #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7