Differentiate polynomial and power functions and interpret a gradient.
Notes and key methods
For a constant real exponent n on a valid differentiable domain, \(\displaystyle \frac{d(x^{n})}{dx}= nx^{n− 1}\). Differentiate a sum term by term; constants have derivative zero. A gradient function gives a different gradient at each point, rather than the equation of the original curve.
At \(\displaystyle x=a\) the tangent uses both f(a) and \(\displaystyle f'(a): y-f(a)=f'(a)(x-a). A\) second derivative is the derivative of the gradient function.
Check the conditions
Respect the domain of fractional or negative powers.
A tangent needs a point as well as a gradient.
Three worked examples
Example 1 · Gradients and the power rule
Let \(f(x)=2 x^{3} + 5 x + 5\), with real x.
Differentiate the given function, showing the rule used.
Find the second derivative.
Find the equation of the tangent at \(x=1\).
Find the y-intercept of the tangent at \(\displaystyle x=1\).
Show the reasoning
(a)
\(6 x^{2} + 5\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 6 x^{2} + 5\)
(b)
Differentiate the complete first derivative and simplify. \(12 x\)
Answer: \(12 x\)
(c)
The point has height \(12\) and gradient \(11\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=11 x + 1\).
Answer: \(y=11 x + 1\)
(d)
Use b=y−mx for the tangent.
\(1\)
Answer: \(1\).
Example 2 · Gradients and the power rule
Let \(f(x)=3 x^{4} + x + 6\), with real x.
Differentiate the given function, showing the rule used.
Find the second derivative.
Find the equation of the tangent at \(x=1\).
Find the y-intercept of the tangent at \(\displaystyle x=1\).
Show the reasoning
(a)
\(12 x^{3} + 1\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 12 x^{3} + 1\)
(b)
Differentiate the complete first derivative and simplify. \(36 x^{2}\)
Answer: \(36 x^{2}\)
(c)
The point has height \(10\) and gradient \(13\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=13 x - 3\).
Answer: \(y=13 x - 3\)
(d)
Use b=y−mx for the tangent.
\(-3\)
Answer: \(-3\).
Example 3 · Gradients and the power rule
Let \(f(x)=4 x^{2} + 2 x + 7\), with real x.
Differentiate the given function, showing the rule used.
Find the second derivative.
Find the equation of the tangent at \(x=1\).
Find the y-intercept of the tangent at \(\displaystyle x=1\).
Show the reasoning
(a)
\(8 x + 2\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 8 x + 2\)
(b)
Differentiate the complete first derivative and simplify. \(8\)
Answer: \(8\)
(c)
The point has height \(13\) and gradient \(10\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=10 x + 3\).