The tangent line to a function at a given point is basically defined as a line which shares the same gradient as that particular given point. Since this function is of a linear nature, it’s gradient is constant for all possible values of x (i.e. the domain). Hence, a line that would share the same gradient as any given point would always have a gradient of -2 (Ignore the person above as the gradient is not zero!). Therefore, the tangent to this particular function would essentially be sitting on the function itself. That is, the tangent to this function is the function itself.

Weird, huh?

In case you’re wondering how to differentiate a polynomial function with powers of any real number.

Let f(x) = ax^(n) + bx^(n-1) + cx^(n-2) + … … …

Then f'(x) = nax^(n-1) + (n-1)bx^(n-2) + (n-2)cx^(n-3) + … … …

See the pattern? You get a term, multiply it by the coefficient of the power, then minus 1 from the power. You repeat for each term in the polynomial. Also, something-to-the-power-of-zero is 1.

eg. Differentiation of 4x^7 is 28x^6

eg. Differentiation of 2x is 2x^0 which is 2 since x^0 is 1

eg. Differentiation of 3-2x is basically a differentiation of 3x^0 – 2x^1 which should yield 0*3*x^(-1) – 1*2*x^0 which is 0 – 2 which is -2.

Good luck and hope you understood that!

f'(x) = -2

it’s a straight line, so the gradient is zero (?)