{"id":135,"date":"2026-03-14T20:14:10","date_gmt":"2026-03-14T20:14:10","guid":{"rendered":"https:\/\/seonumber1.com\/calc\/?page_id=135"},"modified":"2026-03-19T20:27:37","modified_gmt":"2026-03-19T20:27:37","slug":"derivative-calculator","status":"publish","type":"page","link":"https:\/\/seonumber1.com\/calc\/derivative-calculator\/","title":{"rendered":"Derivative Calculator"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"135\" class=\"elementor elementor-135\">\n\t\t\t\t<div class=\"elementor-element elementor-element-320624b e-flex e-con-boxed e-con e-parent\" data-id=\"320624b\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ba9b5b1 elementor-widget elementor-widget-html\" data-id=\"ba9b5b1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link href=\"https:\/\/fonts.googleapis.com\/css2?family=DM+Sans:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n<style>*,*::before,*::after{box-sizing:border-box;margin:0;padding:0}.cw{font-family:'DM Sans',sans-serif;background:#f5f0e8;color:#1a2744;padding:40px 20px;max-width:720px;margin:0 auto}.cw h1{font-size:clamp(1.55rem,3vw,2rem);font-weight:700;text-align:center;margin-bottom:8px}.sub{font-size:.9rem;color:#718096;text-align:center;margin-bottom:28px;line-height:1.6}.tabs{display:flex;gap:6px;flex-wrap:wrap;margin-bottom:20px;justify-content:center}.tab{padding:7px 15px;border:1.5px solid #e2e8f0;border-radius:6px;font-family:inherit;font-size:.8rem;font-weight:600;cursor:pointer;background:#fff;color:#4a5568;transition:all .18s}.tab.on{background:#e8392a;color:#fff;border-color:#e8392a}.cc{background:#fff;border:1px solid #e2e8f0;border-radius:12px;padding:28px;margin-bottom:20px;box-shadow:0 2px 12px rgba(0,0,0,.06)}.cc h2{font-size:.95rem;font-weight:700;color:#1a2744;margin-bottom:16px;padding-bottom:10px;border-bottom:1px solid #f0eae0}.panel{display:none}.panel.on{display:block}.fr{display:grid;grid-template-columns:1fr 1fr;gap:14px;margin-bottom:14px}.fd{display:flex;flex-direction:column;gap:5px}.fd label{font-size:.73rem;font-weight:600;color:#4a5568;letter-spacing:.04em;text-transform:uppercase}.fd input,.fd select{padding:10px 12px;border:1.5px solid #e2e8f0;border-radius:7px;font-family:inherit;font-size:.88rem;color:#1a2744;background:#fafaf8;outline:none;transition:border-color .18s}.fd input:focus,.fd select:focus{border-color:#e8392a;background:#fff}.btn{width:100%;padding:13px;background:#e8392a;color:#fff;font-family:inherit;font-size:.9rem;font-weight:700;border:none;border-radius:8px;cursor:pointer;margin-top:6px;transition:background .18s,transform .15s}.btn:hover{background:#c8301f;transform:translateY(-1px)}.rb{background:#f5f0e8;border:1.5px solid #e8d9c8;border-radius:9px;padding:22px;margin-top:18px;display:none}.rb.show{display:block}.rm{font-size:1.4rem;font-weight:700;color:#e8392a;text-align:center;margin-bottom:4px}.rl{font-size:.73rem;text-transform:uppercase;letter-spacing:.09em;color:#718096;text-align:center;margin-bottom:12px}.steps{font-size:.83rem;color:#4a5568;line-height:1.8;background:#fff;border-radius:8px;padding:14px}.step{padding:5px 0;border-bottom:1px solid #f0eae0}.step:last-child{border:none}.sn{font-weight:700;color:#e8392a;margin-right:6px}.rg{display:grid;grid-template-columns:1fr 1fr;gap:10px;margin-bottom:14px}.ri{background:#fff;border-radius:8px;padding:12px;text-align:center}.ri .rv{font-size:1.1rem;font-weight:700;color:#1a2744}.ri .rll{font-size:.68rem;color:#718096;margin-top:3px}.ib{background:#fff;border:1px solid #e2e8f0;border-radius:12px;padding:22px;box-shadow:0 2px 12px rgba(0,0,0,.06)}.ib h3{font-size:.9rem;font-weight:700;color:#1a2744;margin-bottom:9px}.ib p,.ib li{font-size:.82rem;color:#4a5568;line-height:1.7}.ib ul{padding-left:16px;margin-top:6px}.ib li{margin-bottom:3px}@media(max-width:520px){.fr,.rg{grid-template-columns:1fr}}<\/style>\r\n<div class=\"cw\">\r\n  <h1>d\/dx Derivative Calculator<\/h1>\r\n  <p class=\"sub\">Calculate derivatives analytically and numerically. Find the derivative formula, evaluate at a point, and understand slope and tangent lines.<\/p>\r\n  <div class=\"tabs\">\r\n    <button class=\"tab on\" onclick=\"sw(0)\">Derivative Formula<\/button>\r\n    <button class=\"tab\" onclick=\"sw(1)\">Evaluate at Point<\/button>\r\n    <button class=\"tab\" onclick=\"sw(2)\">Tangent Line<\/button>\r\n    <button class=\"tab\" onclick=\"sw(3)\">Derivative Rules<\/button>\r\n  <\/div>\r\n  <div class=\"cc\">\r\n    <div class=\"panel on\" id=\"p0\">\r\n      <h2>Derivative Formula<\/h2>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Select Function<\/label>\r\n          <select id=\"d_fx\">\r\n            <option value=\"xn\">x\u207f (power rule)<\/option>\r\n            <option value=\"sinx\">sin(x)<\/option><option value=\"cosx\">cos(x)<\/option>\r\n            <option value=\"tanx\">tan(x)<\/option><option value=\"ex\">e\u02e3<\/option>\r\n            <option value=\"lnx\">ln(x)<\/option><option value=\"sqrtx\">\u221ax<\/option>\r\n            <option value=\"x2\">x\u00b2<\/option><option value=\"x3\">x\u00b3<\/option><option value=\"x4\">x\u2074<\/option>\r\n          <\/select>\r\n        <\/div>\r\n        <div class=\"fd\"><label>Power n (for x\u207f only)<\/label><input type=\"number\" id=\"d_n\" placeholder=\"e.g. 5\" step=\"any\" value=\"5\"><\/div>\r\n      <\/div>\r\n      <button class=\"btn\" onclick=\"calc(0)\">Find Derivative<\/button>\r\n      <div class=\"rb\" id=\"r0\">\r\n        <div class=\"rm\" id=\"r0v\">--<\/div>\r\n        <div class=\"rl\">Derivative f'(x)<\/div>\r\n        <div class=\"steps\" id=\"r0s\"><\/div>\r\n      <\/div>\r\n    <\/div>\r\n    <div class=\"panel\" id=\"p1\">\r\n      <h2>Evaluate Derivative at a Point<\/h2>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Function<\/label>\r\n          <select id=\"e_fx\"><option value=\"x2\">x\u00b2<\/option><option value=\"x3\">x\u00b3<\/option><option value=\"sinx\">sin(x)<\/option><option value=\"cosx\">cos(x)<\/option><option value=\"ex\">e\u02e3<\/option><option value=\"lnx\">ln(x)<\/option><option value=\"sqrtx\">\u221ax<\/option><\/select>\r\n        <\/div>\r\n        <div class=\"fd\"><label>Point x = <\/label><input type=\"number\" id=\"e_x\" placeholder=\"e.g. 3\" step=\"any\"><\/div>\r\n      <\/div>\r\n      <button class=\"btn\" onclick=\"calc(1)\">Evaluate f'(x)<\/button>\r\n      <div class=\"rb\" id=\"r1\">\r\n        <div class=\"rg\">\r\n          <div class=\"ri\"><div class=\"rv\" id=\"r1fx\">--<\/div><div class=\"rll\">f(x) value<\/div><\/div>\r\n          <div class=\"ri\"><div class=\"rv\" id=\"r1dfx\">--<\/div><div class=\"rll\">f'(x) value<\/div><\/div>\r\n        <\/div>\r\n        <div class=\"steps\" id=\"r1s\"><\/div>\r\n      <\/div>\r\n    <\/div>\r\n    <div class=\"panel\" id=\"p2\">\r\n      <h2>Tangent Line at a Point<\/h2>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Function<\/label>\r\n          <select id=\"tl_fx\"><option value=\"x2\">x\u00b2<\/option><option value=\"x3\">x\u00b3<\/option><option value=\"sinx\">sin(x)<\/option><option value=\"cosx\">cos(x)<\/option><option value=\"ex\">e\u02e3<\/option><\/select>\r\n        <\/div>\r\n        <div class=\"fd\"><label>Point x = a<\/label><input type=\"number\" id=\"tl_x\" placeholder=\"e.g. 2\" step=\"any\"><\/div>\r\n      <\/div>\r\n      <button class=\"btn\" onclick=\"calc(2)\">Find Tangent Line<\/button>\r\n      <div class=\"rb\" id=\"r2\">\r\n        <div class=\"rm\" id=\"r2v\" style=\"font-size:1.2rem\">--<\/div>\r\n        <div class=\"rl\">Tangent Line Equation<\/div>\r\n        <div class=\"steps\" id=\"r2s\"><\/div>\r\n      <\/div>\r\n    <\/div>\r\n    <div class=\"panel\" id=\"p3\">\r\n      <h2>Differentiation Rules Reference<\/h2>\r\n      <div style=\"font-size:.84rem;color:#4a5568;line-height:2.1\">\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">Power Rule:<\/span> d\/dx [x\u207f] = n\u00b7x\u207f\u207b\u00b9<\/div>\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">Chain Rule:<\/span> d\/dx [f(g(x))] = f'(g(x)) \u00b7 g'(x)<\/div>\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">Product Rule:<\/span> d\/dx [u\u00b7v] = u'v + uv'<\/div>\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">Quotient Rule:<\/span> d\/dx [u\/v] = (u'v \u2212 uv') \/ v\u00b2<\/div>\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">d\/dx [sin x]<\/span> = cos x<\/div>\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">d\/dx [cos x]<\/span> = \u2212sin x<\/div>\r\n        <div style=\"padding:5px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">d\/dx [e\u02e3]<\/span> = e\u02e3<\/div>\r\n        <div style=\"padding:5px 0\"><span style=\"font-weight:700;color:#e8392a\">d\/dx [ln x]<\/span> = 1\/x<\/div>\r\n      <\/div>\r\n    <\/div>\r\n  <\/div>\r\n  <div class=\"ib\"><h3>Numerical Differentiation<\/h3><p>When exact formulas aren't available, use: f'(x) \u2248 [f(x+h) \u2212 f(x\u2212h)] \/ (2h) where h is a small number like 0.0001.<\/p><\/div>\r\n<\/div>\r\n<script>\r\nfunction sw(i){document.querySelectorAll('.tab').forEach(function(t,j){t.classList.toggle('on',j===i)});document.querySelectorAll('.panel').forEach(function(p,j){p.classList.toggle('on',j===i)});}\r\nvar fns={x2:function(x){return x*x;},x3:function(x){return x*x*x;},x4:function(x){return x*x*x*x;},sinx:Math.sin,cosx:Math.cos,tanx:Math.tan,ex:Math.exp,lnx:Math.log,sqrtx:Math.sqrt};\r\nvar dfns={x2:function(x){return 2*x;},x3:function(x){return 3*x*x;},x4:function(x){return 4*x*x*x;},sinx:Math.cos,cosx:function(x){return-Math.sin(x);},tanx:function(x){return 1\/(Math.cos(x)*Math.cos(x));},ex:Math.exp,lnx:function(x){return 1\/x;},sqrtx:function(x){return 1\/(2*Math.sqrt(x));}};\r\nvar deriv={x2:\"2x\",x3:\"3x\u00b2\",x4:\"4x\u00b3\",sinx:\"cos(x)\",cosx:\"\u2212sin(x)\",tanx:\"sec\u00b2(x)\",ex:\"e\u02e3\",lnx:\"1\/x\",sqrtx:\"1\/(2\u221ax)\"};\r\nvar fnlbl={x2:\"x\u00b2\",x3:\"x\u00b3\",x4:\"x\u2074\",sinx:\"sin(x)\",cosx:\"cos(x)\",tanx:\"tan(x)\",ex:\"e\u02e3\",lnx:\"ln(x)\",sqrtx:\"\u221ax\"};\r\nfunction f4(v){return parseFloat(v.toFixed(6));}\r\nfunction calc(m){\r\n  if(m===0){\r\n    var fx=document.getElementById('d_fx').value,n=+document.getElementById('d_n').value;\r\n    var dl=fx==='xn'?n+'x^'+(n-1):(deriv[fx]||'?');\r\n    var fl=fx==='xn'?'x^'+n:fnlbl[fx];\r\n    document.getElementById('r0v').textContent=\"f'(x) = \"+dl;\r\n    document.getElementById('r0s').innerHTML='<div class=\"step\"><span class=\"sn\">f(x) =<\/span> '+fl+'<\/div><div class=\"step\"><span class=\"sn\">Rule:<\/span> '+(fx==='xn'?'Power Rule: d\/dx[x\u207f] = n\u00b7x\u207f\u207b\u00b9':'Standard derivative')+'<\/div><div class=\"step\"><span class=\"sn\">f\\'(x) =<\/span> '+dl+'<\/div>';\r\n    document.getElementById('r0').classList.add('show');\r\n  } else if(m===1){\r\n    var fx=document.getElementById('e_fx').value,x=+document.getElementById('e_x').value;\r\n    if(isNaN(x)){alert('Enter a value for x.');return;}\r\n    var fv=f4(fns[fx](x)),dfv=f4(dfns[fx](x));\r\n    document.getElementById('r1fx').textContent=fv;document.getElementById('r1dfx').textContent=dfv;\r\n    document.getElementById('r1s').innerHTML='<div class=\"step\"><span class=\"sn\">f(x) =<\/span> '+fnlbl[fx]+'<\/div><div class=\"step\"><span class=\"sn\">f\\'(x) =<\/span> '+deriv[fx]+'<\/div><div class=\"step\"><span class=\"sn\">At x =<\/span> '+x+'<\/div><div class=\"step\"><span class=\"sn\">f('+x+') =<\/span> '+fv+'<\/div><div class=\"step\"><span class=\"sn\">f\\'('+x+') =<\/span> '+dfv+' \u2190 slope at this point<\/div>';\r\n    document.getElementById('r1').classList.add('show');\r\n  } else {\r\n    var fx=document.getElementById('tl_fx').value,a=+document.getElementById('tl_x').value;\r\n    if(isNaN(a)){alert('Enter a value for x.');return;}\r\n    var ya=f4(fns[fx](a)),m2=f4(dfns[fx](a)),b=f4(ya-m2*a);\r\n    var eq='y = '+m2+'x '+(b>=0?'+ '+b:'\u2212 '+Math.abs(b));\r\n    document.getElementById('r2v').textContent=eq;\r\n    document.getElementById('r2s').innerHTML='<div class=\"step\"><span class=\"sn\">f(x) =<\/span> '+fnlbl[fx]+'<\/div><div class=\"step\"><span class=\"sn\">Point:<\/span> ('+a+', '+ya+')<\/div><div class=\"step\"><span class=\"sn\">Slope m = f\\'('+a+') =<\/span> '+m2+'<\/div><div class=\"step\"><span class=\"sn\">y \u2212 '+ya+' = '+m2+'(x \u2212 '+a+')<\/span><\/div><div class=\"step\"><span class=\"sn\">Tangent line:<\/span> '+eq+'<\/div>';\r\n    document.getElementById('r2').classList.add('show');\r\n  }\r\n}\r\n<\/script>\r\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-d1f77f8 e-flex e-con-boxed e-con e-parent\" data-id=\"d1f77f8\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b6bf487 elementor-widget elementor-widget-text-editor\" data-id=\"b6bf487\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h3 class=\"text-xl font-semibold text-foreground mb-4 flex items-center\">How the\u00a0Derivative Calculator\u00a0Works<\/h3><div class=\"space-y-6\"><div><h4 class=\"font-semibold mb-3 text-lg\">How Derivatives Work<\/h4><p class=\"text-sm mb-4 text-muted-foreground\">A derivative measures the instantaneous rate of change of a function at a specific point. It represents the slope of the tangent line to the function&#8217;s graph at that point.<\/p><\/div><div><h4 class=\"font-semibold mb-2\">Mathematical Formulas:<\/h4><div class=\"space-y-3 text-sm\"><div class=\"bg-secondary\/10 p-3 rounded\"><strong>Definition (Limit):<\/strong><br \/>f'(x) = lim[h\u21920] [f(x+h) &#8211; f(x)] \/ h<br \/><em class=\"text-muted-foreground\">Fundamental definition of derivative<\/em><\/div><div class=\"bg-secondary\/10 p-3 rounded\"><strong>Power Rule:<\/strong><br \/>If f(x) = x\u207f, then f'(x) = n\u00b7x\u207f\u207b\u00b9<br \/><em class=\"text-muted-foreground\">Most common differentiation rule<\/em><\/div><div class=\"bg-secondary\/10 p-3 rounded\"><strong>Numerical Approximation:<\/strong><br \/>f'(x) \u2248 [f(x+h) &#8211; f(x-h)] \/ (2h)<br \/><em class=\"text-muted-foreground\">Central difference method<\/em><\/div><\/div><\/div><div><h4 class=\"font-semibold mb-2\">Real-World Applications:<\/h4><ul class=\"list-disc ml-6 space-y-1 text-sm text-muted-foreground\"><li><strong>Physics:<\/strong>\u00a0Velocity (derivative of position), acceleration (derivative of velocity)<\/li><li><strong>Economics:<\/strong>\u00a0Marginal cost, marginal revenue, elasticity of demand<\/li><li><strong>Engineering:<\/strong>\u00a0Rate of heat transfer, stress analysis, optimization<\/li><li><strong>Biology:<\/strong>\u00a0Population growth rates, reaction rates in biochemistry<\/li><li><strong>Finance:<\/strong>\u00a0Option pricing models, risk analysis, portfolio optimization<\/li><\/ul><\/div><div><h4 class=\"font-semibold mb-2\">Usage Recommendations:<\/h4><ul class=\"list-disc ml-6 space-y-1 text-sm text-muted-foreground\"><li>Use analytical method for exact results with simple functions<\/li><li>Choose numerical method for complex functions or when exact form is unknown<\/li><li>For optimization problems, find where derivative equals zero (critical points)<\/li><li>Use second derivative test to determine if critical points are maxima or minima<\/li><li>For numerical methods, use smaller step sizes for higher accuracy<\/li><\/ul><\/div><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>d\/dx Derivative Calculator Calculate derivatives analytically and numerically. 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