Differentiation Using the Chain Rule SOLUTION 1 : Differentiate. \(\displaystyle y=\cos \left( {4x} \right)\), \(\displaystyle g\left( x \right)=\cos \left( {\tan x} \right)\), \(\displaystyle \begin{array}{l}f\left( x \right)={{\sec }^{3}}\left( {\pi x} \right)\\f\left( x \right)={{\left[ {\sec \left( {\pi x} \right)} \right]}^{3}}\end{array}\), \(\displaystyle \begin{array}{l}f\left( \theta \right)=2{{\cot }^{2}}\left( {2\theta } \right)+\theta \\f\left( \theta \right)=2{{\left[ {\cot \left( {2\theta } \right)} \right]}^{2}}+\theta \end{array}\). Evaluate any superscripted expression down to a single number before evaluating the power. In the next section, we use the Chain Rule to justify another differentiation technique. Example 1 Find the derivative f '(x), if f is given by f(x) = 4 cos (5x - 2) Solution to Example 1 Let u = 5x - 2 and f(u) = 4 cos u, hence du / dx = 5 and df / du = - 4 sin u We now use the chain rule eval(ez_write_tag([[580,400],'shelovesmath_com-medrectangle-4','ezslot_2',110,'0','0']));Understand these problems, and practice, practice, practice! An expression in an exponent (a small, raised number indicating a power) groups that expression like parentheses do. Yes, sometimes we have to use the chain rule twice, in the cases where we have a function inside a function inside another function. Anytime there is a parentheses followed by an exponent is the general rule of thumb. Use the Product Rule, since we have \(t\)âs in both expressions. \(\displaystyle \begin{align}{l}{g}â\left( x \right)&=\frac{1}{4}{{\left( {\color{red}{{16-{{x}^{3}}}}} \right)}^{{-\frac{3}{4}}}}\cdot \left( {\color{red}{{-3{{x}^{2}}}}} \right)\\&=-\frac{{3{{x}^{2}}}}{{4{{{\left( {16-{{x}^{3}}} \right)}}^{{\frac{3}{4}}}}}}=-\frac{{3{{x}^{2}}}}{{4\,\sqrt[4]{{{{{\left( {16-{{x}^{3}}} \right)}}^{3}}}}}}\end{align}\). When should you use the Chain Rule? 4. The reason we also took out a \(\frac{3}{2}\) is because itâs the GCF of \(\frac{3}{2}\) and \(\frac{{24}}{2}\,\,(12)\). \(\displaystyle \begin{align}{f}â\left( x \right)&=3\,{{\color{red}{{\sec }}}^{2}}\left( {\color{blue}{{\pi x}}} \right)\cdot \left( {\color{red}{{\sec \left( {\color{blue}{{\pi x}}} \right)\tan \left( {\color{blue}{{\pi x}}} \right)}}} \right)\color{blue}{\pi }\\&=3\pi {{\sec }^{3}}\left( {\pi x} \right)\tan \left( {\pi x} \right)\end{align}\), This oneâs a little tricky, since we have to use the Chain Rule, \(\displaystyle \begin{align}{f}â\left( \theta \right)=&4\,\color{red}{{\cot }}\left( {\color{blue}{{2\theta }}} \right)\cdot \color{red}{{-{{{\csc }}^{2}}\left( {\color{blue}{{2\theta }}} \right)}}\cdot \color{blue}{2}+1\\&=1-8{{\csc }^{2}}\left( {2\theta } \right)\cot \left( {2\theta } \right)\end{align}\). Since the last step is multiplication, we treat the express Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another and then both of them are inside a third function (you can have four or more nested functions, but three is probably the most you’ll see). This is more formally stated as, if the functions f (x) and g (x) are both differentiable and define F (x) = (f o g)(x), then the required derivative of the function F(x) is, This formal approach … ... To evaluate the expression above you (1) evaluate the expression inside the parentheses and the (2) raise that result to the 53 power. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Browse other questions tagged derivatives chain-rule transcendental-equations or ask your own question. \(\displaystyle \begin{array}{l}{y}â=-\sin \left( {\color{red}{{4x}}} \right)\cdot \color{red}{4}\\=-4\sin \left( {4x} \right)\end{array}\), Since the \(\left( {4x} \right)\) is the inner function (the argument of \(\text{sin}\)), we have to take multiply by the derivative of that function, which is, \(\displaystyle \begin{align}{g}â\left( x \right)&=-\sin \left( {\color{red}{{\tan x}}} \right)\cdot \color{red}{{{{{\sec }}^{2}}x}}\\&=-{{\sec }^{2}}x\cdot \sin \left( {\tan x} \right)\end{align}\). The equation of the tangent line to \(f\left( \theta \right)=\cos \left( {5\theta } \right)\) at the point \(\displaystyle \left( {\frac{\pi }{2},0} \right)\) is \(\displaystyle y=-5x+\frac{{5\pi }}{2}\). are the inner functions, we have to multiply each by their derivative. The chain rule says when weâre taking the derivative, if thereâs something other than \(\boldsymbol {x}\) (like in parentheses or under a radical sign) when weâre using one of the rules weâve learned (like the power rule), we have to multiply by the derivative of whatâs in the parentheses. \(\begin{array}{c}f\left( x \right)={{\left( {5{{x}^{4}}-2} \right)}^{3}}\\x=1\end{array}\), \(\displaystyle {f}â\left( x \right)=3{{\left( {5{{x}^{4}}-2} \right)}^{2}}\left( {20{{x}^{3}}} \right)=60{{x}^{3}}{{\left( {5{{x}^{4}}-2} \right)}^{2}}\). As an example, let's analyze 4•(x³+5)² Speaking informally we could say the "inside function" is (x 3 +5) and the "outside function" is 4 • (inside) 2. Take a look at the same example listed above. 312. f (x) = (2 x3 + 1) (x5 – x) The chain rule is a rule, in which the composition of functions is differentiable. \({p}â\left( 4 \right)\text{ and }{q}â\left( {-1} \right)\), The Equation of the Tangent Line with the Chain Rule, \(\displaystyle \begin{align}{f}â\left( x \right)&=8{{\left( {\color{red}{{5x-1}}} \right)}^{7}}\cdot \color{red}{5}\\&=40{{\left( {5x-1} \right)}^{7}}\end{align}\), Since the \(\left( {5x-1} \right)\) is the inner function, after using the Power Rule, we have to multiply by the derivative of that function, which is, \(\displaystyle \begin{align}{f}â\left( x \right)&=3{{\left( {\color{red}{{{{x}^{4}}-1}}} \right)}^{2}}\cdot \left( {\color{red}{{4{{x}^{3}}}}} \right)\\&=12{{x}^{3}}{{\left( {{{x}^{4}}-1} \right)}^{2}}\end{align}\). So let’s dive right into it! At point \(\left( {1,27} \right)\), the slope is \(\displaystyle 60{{\left( 1 \right)}^{3}}{{\left[ {5{{{\left( 1 \right)}}^{4}}-2} \right]}^{2}}=540\). Since the \(\left( {{{x}^{4}}-1} \right)\) is the inner function, after using the Power Rule, we have to multiply by the derivative of that function, which is \(4{{x}^{3}}\). So use your parentheses! And part of the reason is that students often forget to use it when they should. Sometimes, you'll use it when you don't see parentheses but they're implied. Section 2.5 The Chain Rule. Then when the value of g changes by an amount Δg, the value of f will change by an amount Δf. I must say I'm really surprised not one of the answers mentions that. ;
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