Derivatives - Product Rule For Derivatives

Derivatives - Product Rule for Derivatives

Introduction

The product rule is a formula used to find the derivative of a function that is the product of two other functions.

Formula

If we have two functions, f(x) and g(x), then the product rule states: (f * g)' = f' * g + f * g'

Explanation

The derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.

Example 1

Find the derivative of the function f(x) = x^2 * sin(x). Step 1: Identify f(x) and g(x).

f(x) = x^2
g(x) = sin(x) Step 2: Find f’(x) and g’(x).
f’(x) = 2x
g’(x) = cos(x) Step 3: Apply the product rule formula. (f * g)' = f' * g + f * g' (x^2 * sin(x))' = (2x * sin(x)) + (x^2 * cos(x))

Example 2

Find the derivative of the function f(x) = e^x * ln(x). Step 1: Identify f(x) and g(x).

f(x) = e^x
g(x) = ln(x) Step 2: Find f’(x) and g’(x).
f’(x) = e^x
g’(x) = 1/x Step 3: Apply the product rule formula. (f * g)' = f' * g + f * g' (e^x * ln(x))' = (e^x * 1/x) + (e^x * ln(x))

Derivative Rules Reminder

The power rule: (x^n)' = nx^(n-1)
The constant rule: (c)' = 0
The sum rule: (f + g)' = f' + g'

Conclusion

The product rule allows us to find the derivative of a function that is the product of two other functions. By using the formula (f * g)' = f' * g + f * g', we can differentiate such functions efficiently and accurately.

</p>
</section>
<section style="font-size: 50px";>
<h2 id="derivatives-of-trigonometric-functions">Derivatives of Trigonometric Functions</h2>
</section>
<section style="font-size: 50px";>
<h2 id="derivative-of-sinx">Derivative of sin(x)</h2>
<ul>
<li>The derivative of <code>sin(x)</code> is <code>cos(x)</code>.</li>
<li>Example: <code>(sin(x))' = cos(x)</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="derivative-of-cosx">Derivative of cos(x)</h2>
<ul>
<li>The derivative of <code>cos(x)</code> is <code>-sin(x)</code>.</li>
<li>Example: <code>(cos(x))' = -sin(x)</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="derivative-of-tanx">Derivative of tan(x)</h2>
<ul>
<li>The derivative of <code>tan(x)</code> is <code>sec^2(x)</code>.</li>
<li>Example: <code>(tan(x))' = sec^2(x)</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="derivative-of-cscx">Derivative of csc(x)</h2>
<ul>
<li>The derivative of <code>csc(x)</code> is <code>-csc(x) * cot(x)</code>.</li>
<li>Example: <code>(csc(x))' = -csc(x) * cot(x)</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="derivative-of-secx">Derivative of sec(x)</h2>
<ul>
<li>The derivative of <code>sec(x)</code> is <code>sec(x) * tan(x)</code>.</li>
<li>Example: <code>(sec(x))' = sec(x) * tan(x)</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="chain-rule-for-derivatives">Chain Rule for Derivatives</h2>
</section>
<section style="font-size: 50px";>
<h2 id="formula-1">Formula</h2>
<ul>
<li>The chain rule allows us to find the derivative of composite functions.</li>
<li>If <code>f(g(x))</code> is a composite function, then its derivative is given by <code>f'(g(x)) * g'(x)</code>.</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="example-1-1">Example 1</h2>
<p>Find the derivative of the function <code>f(x) = sin(2x)</code>.
Step 1: Identify f(x) and g(x).</p>
<ul>
<li>f(x) = sin(x)</li>
<li>g(x) = 2x
Step 2: Find f’(x) and g’(x).</li>
<li>f’(x) = cos(x)</li>
<li>g’(x) = 2
Step 3: Apply the chain rule formula.
<code>(f(g(x)))' = f'(g(x)) * g'(x)</code>
<code>(sin(2x))' = cos(2x) * 2</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="example-2-1">Example 2</h2>
<p>Find the derivative of the function <code>f(x) = e^(3x)</code>.
Step 1: Identify f(x) and g(x).</p>
<ul>
<li>f(x) = e^x</li>
<li>g(x) = 3x
Step 2: Find f’(x) and g’(x).</li>
<li>f’(x) = e^x</li>
<li>g’(x) = 3
Step 3: Apply the chain rule formula.
<code>(f(g(x)))' = f'(g(x)) * g'(x)</code>
<code>(e^(3x))' = e^(3x) * 3</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="chain-rule-with-multiple-functions">Chain Rule with Multiple Functions</h2>
<ul>
<li>The chain rule can be applied to functions with multiple nested functions.</li>
<li>Example: <code>f(g(h(x)))</code></li>
<li>The derivative would be <code>f'(g(h(x))) * g'(h(x)) * h'(x)</code></li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="conclusion-1">Conclusion</h2>
<ul>
<li>Understanding the derivatives of trigonometric functions and the chain rule is important for solving complex problems in calculus.</li>
<li>The derivative rules and formulas discussed in these slides can be applied to a wide range of mathematical functions and help in finding rates of change and optimizing functions.</li>
<li>Regular practice and problem-solving can further enhance your understanding and proficiency in applying these concepts.</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slides-21-30">Slides 21-30</h2>
</section>
<section style="font-size: 50px";>
<h2 id="slide-21">Slide 21</h2>
<ul>
<li>Applications of Integration
<ul>
<li>Area under a curve</li>
<li>Volume of revolution</li>
<li>Finding the center of mass</li>
<li>Determining arc length</li>
<li>Solving differential equations</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-22">Slide 22</h2>
<ul>
<li>Area under a curve
<ul>
<li>Integrating a function gives the area between the curve and the x-axis.
Example: Find the area under the curve <code>y = sin(x)</code> from <code>x = 0</code> to <code>x = π</code>.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-23">Slide 23</h2>
<ul>
<li>Volume of revolution
<ul>
<li>Integrating a function multiplied by its radius squared gives the volume of the solid formed by rotating the curve about an axis.
Example: Find the volume of the solid formed by rotating the curve <code>y = x^2</code> about the x-axis from <code>x = 0</code> to <code>x = 2</code>.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-24">Slide 24</h2>
<ul>
<li>Finding the center of mass
<ul>
<li>Integrating a function multiplied by its distance from an axis gives the weighted average position of the mass.
Example: Find the x-coordinate of the center of mass of the region bounded by the curve <code>y = x^2</code> and the x-axis.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-25">Slide 25</h2>
<ul>
<li>Determining arc length
<ul>
<li>Integrating the square root of the sum of the squares of the derivatives gives the length of a curve.
Example: Find the length of the curve <code>y = e^x</code> from <code>x = 0</code> to <code>x = 1</code>.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-26">Slide 26</h2>
<ul>
<li>Solving differential equations
<ul>
<li>Integration plays a crucial role in solving differential equations, which model various processes in science and engineering.
Example: Solve the differential equation <code>y' = 2x</code> with the initial condition <code>y(0) = 1</code>.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-27">Slide 27</h2>
<ul>
<li>Integration by Parts
<ul>
<li>Another technique to evaluate integrals is integration by parts.</li>
<li>It is based on the product rule of differentiation.
Formula: ∫ u dv = u v - ∫ v du</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-28">Slide 28</h2>
<ul>
<li>Example: Evaluating an integral using integration by parts
∫ x ln(x) dx
Step 1: Choose u and dv
u = ln(x)    dv = x dx</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-29">Slide 29</h2>
<ul>
<li>Example: Evaluating an integral using integration by parts
Step 2: Find du and v
du = (1/x) dx    v = x^2/2
Step 3: Apply the integration by parts formula
∫ x ln(x) dx = (x^2/2) ln(x) - ∫ (x^2/2) (1/x) dx</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-30">Slide 30</h2>
<ul>
<li>Example: Evaluating an integral using integration by parts
Step 4: Simplify the right-hand side
∫ x ln(x) dx = (x^2/2) ln(x) - ∫ (x/2) dx
= (x^2/2) ln(x) - (x^2/4) + C
where C is the constant of integration.</li>
</ul>

</section>

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