Derivatives review

Determine the instantaneous gradient/rate of change of the curve with equation $$y=x^2-4x+5$$ at the following points. (a) (2,1) (b) (-1,10)

The tangent to the curve with equation $$y=x^3+3x^2-2x$$ at the point A has gradient 22. Given that x>0 for the point A, determine the x-coordinate of A.

A curve has equation $$y=2x^2+3x-9$$ 9. At a point P on the curve, the tangent is parallel to the line y=4−5x . What are the coordinates of P?

A curve with equation y=f(x) passes through the point (4,25). f(x)= $$\frac{1}{8}x^3-20x^{\frac{1}{2}}+x+53$$ where x≥0 . Find an equation of the normal to the curve at the point (4,25).

The point P on the curve $$y=6\sqrt[]{x}$$ has x-coordinate 4. The normal to the curve at P is parallel to the line $$2x+3y=0$$ . This normal meets the x-axis at the point Q. Calculate the area of the triangle OPQ, where O is the point (0,0).

Determine the second derivative for each function. $$y=x^7−3x^6+2$$

Determine the second derivative of the given function at the specific point. $$f\left(x\right)=\frac{4+x^2}{2\sqrt[]{x}}$$ at (4,5)

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