The derivative f (x) gives the tangent gradient where the curve is differentiable. dydx=f (x) ddx(xn)=nxn-1 Apply the power rule where the function and derivative exist. Rewrite roots and reciprocals as powers first. For a straight line, the gradient is y_2-y_1x_2-x_1: vertical change divided by horizontal change. Example For y=3, the gradient is 0. For both y=2x and y=2x-3, the gradient is 2. For y=x2, the gradient depends on the point on the curve. dydx =2x At x=-1,0,1,2, the gradients are respectively -2,0,2,4. At x=a, the gradient is 2a.

The derivative \(\mathrm{f}^{\prime}(x)\) gives the tangent gradient where the curve is differentiable.

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{f}^{\prime}(x)\qquad\dfrac{\mathrm{d}}{\mathrm{d}x}(x^n)=nx^{n-1}\)

Apply the power rule where the function and derivative exist. Rewrite roots and reciprocals as powers first.

For a straight line, the gradient is \(\dfrac{y_2-y_1}{x_2-x_1}\): vertical change divided by horizontal change.

For \(y=3\), the gradient is \(0\). For both \(y=2x\) and \(y=2x-3\), the gradient is \(2\).

For \(y=x^2\), the gradient depends on the point on the curve.

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)\({}=2x\)

At \(x=-1,0,1,2\), the gradients are respectively \(-2,0,2,4\). At \(x=a\), the gradient is \(2a\).

Derivative as a Gradient Function and the Power Rule

Derivative as a Gradient Function and the Power Rule

Video Solution: