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<p>Let&nbsp;$\mathrm{f}(x)=(x^2+2x)\mathrm{e}^x$.</p>
<ol class="abc-list"><li data-list-item-id="quadraticexp-part-a" data-list-item-path="quadraticexp-part-a"><div class="list-item-paragraph">Using the definition $\mathrm{g}'(x)=\lim_{h\to0}\dfrac{\mathrm{g}(x+h)-\mathrm{g}(x)}{h}$, show from first principles that the derivative of $\mathrm{g}(x)=x^2+2x$ is $2x+2$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="quadraticexp-part-b" data-list-item-path="quadraticexp-part-b"><div class="list-item-paragraph">Prove by induction that, for every positive integer $n$, the $n^{\mathrm{th}}$ derivative of $\mathrm{f}$ is $\mathrm{f}^{(n)}(x)=\mathrm{e}^x[x^2+2(n+1)x+n(n+1)]$.<span class="inline-marks">[9]</span></div></li></ol>

Let $\mathrm{f}(x)=(x^2+2x)\mathrm{e}^x$.

Using the definition $\mathrm{g}'(x)=\lim_{h\to0}\dfrac{\mathrm{g}(x+h)-\mathrm{g}(x)}{h}$, show from first principles that the derivative of $\mathrm{g}(x)=x^2+2x$ is $2x+2$.

Prove by induction that, for every positive integer $n$, the $n^{\mathrm{th}}$ derivative of $\mathrm{f}$ is $\mathrm{f}^{(n)}(x)=\mathrm{e}^x[x^2+2(n+1)x+n(n+1)]$.

Proof by Mathematical Induction

Differentiation by First Principles

Second and Higher Derivatives

Derivatives of Exponential Functions

Product, Quotient and Chain Rules

Quadratic exponential derivative pattern

Mathematical Induction

Techniques of Differentiation

<ol class="abc-list"><li data-list-item-id="ib2012mhp1tz2q13-part-a" data-list-item-path="ib2012mhp1tz2q13-part-a"><div class="list-item-paragraph">Using the definition of a derivative as&nbsp;$\mathrm{f}'(x)=\lim_{h\to0}\left(\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}\right)$, show that the derivative of $\frac{1}{2x+1}$ is $\frac{-2}{(2x+1)^2}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ib2012mhp1tz2q13-part-b" data-list-item-path="ib2012mhp1tz2q13-part-b"><div class="list-item-paragraph">Prove by induction that the $n^{\mathrm{th}}$ derivative of $(2x+1)^{-1}$ is $(-1)^n\frac{2^n n!}{(2x+1)^{n+1}}$.<span class="inline-marks">[9]</span></div></li></ol>

Using the definition of a derivative as $\mathrm{f}'(x)=\lim_{h\to0}\left(\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}\right)$, show that the derivative of $\frac{1}{2x+1}$ is $\frac{-2}{(2x+1)^2}$.

Prove by induction that the $n^{\mathrm{th}}$ derivative of $(2x+1)^{-1}$ is $(-1)^n\frac{2^n n!}{(2x+1)^{n+1}}$.

IB Mathematics HL May 2012 P1 TZ2 Q13

<p>Let&nbsp;$f(x) = x^3 - 4x +1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Expand $ (x + h)^3$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Use the formula $f'(x) = \lim\limits_{h \to 0} \frac{f(x +h) - f(x)}{h}$ to show that the derivative of $f(x)$ is $3x^2 - 4$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The tangent to the curve of $f$ at the point $ \mathrm{P} (1,-2)$ is parallel to the tangent at a point $\mathrm{Q}$.<br>Find the coordinates of $\mathrm{Q}$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The graph of $f$ is decreasing for&nbsp;$p &lt; x&lt; q$. Find the value of $p$ and of $q$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down the range of values for the gradient of $f$.<span class="inline-marks">[2]</span></div></li></ol>

Binomial Expansion for Positive Integer Powers

Tangents and Normals at a Point

Increasing and Decreasing Functions

Binomial Theorem

Applications of Differentiation

Differentiation

<p>Find the gradient function of the curve $\mathrm{f}\left( x \right)={{x}^{3}}$ from the first principles.</p>

Find the gradient function of the curve $\mathrm{f}\left( x \right)={{x}^{3}}$ from the first principles.

Worked Example 1

<p>Find the gradient function of the following curves from the first principles.</p>
<ol class="i-list"><li><div class="list-item-paragraph">$y={{x}^{3}}$ 	</div></li><li><div class="list-item-paragraph">$y=3{{x}^{2}}$</div></li><li><div class="list-item-paragraph">$y={{\left( x+1 \right)}^{2}}$</div></li><li><div class="list-item-paragraph">$y=\frac{1}{x}$  </div></li></ol>

Find the gradient function of the following curves from the first principles.

For two points $(a,f(a))$ and $(a+\delta x,f(a+\delta x))$, the secant gradient is $m=\frac{f(a+\delta x)-f(a)}{\delta x}$.

As $\delta x\to0$, the secant approaches the tangent, so $f'(a)=\lim_{\delta x\to0}\frac{f(a+\delta x)-f(a)}{\delta x}$.

For $f(x)=x^3$, $f'(x)=\lim_{\delta x\to0}\frac{(x+\delta x)^3-x^3}{\delta x}$.

Expanding and cancelling gives $f'(x)=\lim_{\delta x\to0}\left(3x^2+3x\delta x+(\delta x)^2\right)$.

Therefore $f'(x)=3x^2$, so $\frac{\mathrm d}{\mathrm dx}(x^3)=3x^2$.

First-Principles Derivation

<p>The parametric equations of a curve are $x=at$, $y=a{{t}^{2}}$ where $a$ is a positive constant. The points $P\left( ap,\,\,a{{p}^{2}} \right)$ and $Q\left( aq,\,\,a{{q}^{2}} \right)$ lie on the curve. Find, and simplify an expression, in terms of $p$ and $q$, for the gradient of the chord $PQ$. Deduce, from this expression, the gradient of the curve at the point $Q$.</p>

The parametric equations of a curve are $x=at$, $y=a{{t}^{2}}$ where $a$ is a positive constant. The points $P\left( ap,\,\,a{{p}^{2}} \right)$ and $Q\left( aq,\,\,a{{q}^{2}} \right)$ lie on the curve. Find, and simplify an expression, in terms of $p$ and $q$, for the gradient of the chord $PQ$. Deduce, from this expression, the gradient of the curve at the point $Q$.

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